Trigonometric Functions Questions

We provide trigonometric functions practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on trigonometric functions skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

Trigonometric Functions Questions

List of trigonometric functions Questions

Question NoQuestionsClass
1ze
+cos 20+ cos
+…+cos (3)
10. Let f(0) =
7e
sin – +sin 20+ sin —
+…+sin (3n-2)
then
C.11 (= (2+v3) d. none of these
d. none of these
11
28. The minimum value of
(3 sin x – 4cos X – 10)(3 sin x + 4 cos x – 10)
11
322. If 0 < x < 21 and cosx | < sinx, then
then
a. the set of all values of x is
b. the number of solutions that are integral multiple of

is four
c. the sum of the largest and the smallest solution is to
TTC IT 37
d. the set of all values of x is xe
2
11
4If ( sin theta-sqrt{6} cos theta=sqrt{7} cos theta . ) Prove that
( cos theta-sqrt{6} sin theta-sqrt{7} sin theta=1 )
11
59. Let a, b, and y be some angles in the first quadrant
satisfying tan(a + B) = 15/8 and cosec y = 17/8, then
which of the following hold(s) good?
a. a + B+ y = 11
b. cot a cot ßcot y=cot a + cotß+ coty
c. tan a + tanß+ tan y = tan a tan ſ tan y
d. tan a tan ß + tan ſ tan y+ tan y tan a=1
11
6Find the value of ( frac{sin x}{1+cos x} ) at ( x=frac{pi}{4} )11
7( sum_{k=1}^{6}left[sin frac{2 k pi}{7}-i cos frac{2 k pi}{7}right]= )
A . -1
B.
( c cdot-i )
D.
11
832. If sin 0, – sin 02 = a and cos 6, + cos 02 = b, then
a. a? + b2 24
c. a² +6²23 d. a² +6²52
b. c? + b2 = 4
svart
11
920. The value of cosec
18
.
11
10Prove the following identities. ( sin h(-x)=-sin h x )11
1169. If tan 15° = 2 – 73, the value of
tan 15° cot 75° + tan 75° cot 15°
is
(2) 12
(1) 14
(3) 10
(4) 8
11
12If ( boldsymbol{f}(sin 2 boldsymbol{x})= )
( frac{left(2 tan x+sec ^{2} xright)(1+cos 2 c)}{2}, ) then
determine the range of ( boldsymbol{f}(boldsymbol{t}) ) if range is ( [a, b], ) then ( b=? )
11
1371. The value of sin 15° is:
22
(4) 12+ 45
750 is
11
14If ( frac{cos (boldsymbol{A}-boldsymbol{B})}{cos (boldsymbol{A}+boldsymbol{B})}+frac{cos (boldsymbol{C}+boldsymbol{D})}{cos (boldsymbol{C}-boldsymbol{D})}=mathbf{0}, ) then
( tan A tan B tan C= )
( mathbf{A} cdot tan D )
B. ( cot D )
( c .-tan D )
D. – cot ( D )
11
1516.
In a triangle PQR, ZR=1t/2.Iftan (P/2) and tan (0/2) ar
the roots of the equation ax² +bx+c=0 (a + 0) then.
(1999 – 2 Marks)
(a) a+b=c
(b) b+c=a
(c) a+c=b
(d) b=c
11
16Prove that ( frac{tan mathbf{A}}{mathbf{1 + operatorname { s e c } A}}-frac{tan A}{mathbf{1 – operatorname { s e c } A}}= )
( 2 operatorname{cosec} A )
11
1790. If no solution of 3 sin y + 12 sin’x = a lies on the line
y = 3x, then
a. a € (-o,-9) U (9,-)
b. a e [-9,9]
c. ae {-9,9}
d. none of these
11
18m
1. If tan a =-
and tan ß =
1, find the possible
m
+1
values of (a + B).
(IIT-JEE 1978)
11
1953. One of the general solutions of V3 cos 0 – 3 sin 0 = 4
sin 20 cos 30 is
a. mn + /18, me Z b. mtt/2 + 7t/6, y me Z
11
20Domain of ( sin (cos theta) ) is ( ldots ldots )
A ( cdotleft[-frac{pi}{2}, frac{pi}{2}right] )
в. ( R )
c. ( [0 . pi] )
D cdot [-1,1]
11
2111. The maximum value of y=
sinº x + cos x’
11
22( frac{cos theta}{sin theta} times frac{tan theta}{csc theta} )11
23Find the radian measures
corresponding to ( 5^{0} 37^{prime} 0^{prime prime} )
11
243. in a triangle ABC, 2C –
3. In a triangle ABC, ZC =
Iftan (1) and tan ()
tan
and tan
are the roots of the equation ax² + bx + c = 0 (a + 0),
then the value of a+b (where, a, b, c are sides
of A opposite to angles A, B, C, respectively) is
11
25Illustration 3.105 In AABC
O Prove that cos? 4 + cos2 +cose
WIt cos? At.co tcost = y(x2 + 4) then find the
maximum value of y.
11
26If ( tan theta+tan 4 theta+tan 7 theta= )
( tan theta tan 4 theta tan 7 theta, ) then ( theta= )
A ( cdot frac{n pi}{4} )
в. ( frac{n pi}{7} )
c. ( frac{n pi}{12} )
D. ( n pi )
11
27General solution of equation ( cot theta+ ) ( operatorname{cosec} theta=sqrt{3} ) is
A ( cdot 2 n pi+frac{pi}{4} )
B . ( (2 n-1) pi )
c. ( 2 n pi+frac{pi}{3} )
D. ( 2 n pi+frac{pi}{6} )
11
28Prove that: ( sum sin ^{4} frac{r pi}{16}=frac{3}{2}, r= )
1,3,5,7
11
291. If
+ B+Y = 2੮, then
a tan tan tan ਨੂੰ = ਨੂੰ n en ?
tan
+
tan

c tan ਨੂੰ +tan : +tan ਨੂੰ –tan tan , un ਨੂੰ
d. none of these
(IIT-JEE 1979)
11
30The area of a sector of a circle of radius
( mathbf{7} mathrm{cm} ) and central angle ( 120^{circ} ) is
A ( cdot 152 mathrm{cm}^{2} )
B. ( frac{154}{3} mathrm{cm}^{2} )
c. ( frac{128}{3} mathrm{cm}^{2} )
D. ( 128 mathrm{cm}^{2} )
11
31Let ( [x] ) be the greatest integar function. Then the equation ( sin x=[1+sin x]+[1- )
( cos x] ) has
A . one solution in ( left[frac{-pi}{2}, frac{pi}{2}right] )
B. one solution in ( left[frac{pi}{2}, piright] )
c. one solution in R
D. no solution in R
11
329. cos(a-B) = 1 and cos(a+B) = 1/e, where a, ße [- TT,
Number of pairs of a, ß which satisfy both the equations
is
2.0
a. 0
c. 2
b. 1
d. 4
t
(IT-JEE 2009
(IIT-JEE 2005)
11
33Write ( tan theta ) in terms of ( sin theta )11
34Express as product :
( sin 6 x-sin 2 x )
11
3537. If sin e, sin 02 – cose, cos e, + 1 = 0, then the value of
tan(0/2)cot(02/2) is equal to
b. 120
c. 2
d. – 2
a. – 1
11
362. The set of all x in the interval [0, a) for which 2 sin’x –
3 sin x + 1 20 is
(IIT-JEE 1987)
11
37tan a + tan y
1 + tan tan
prove that
Illustration 3.63 If tan ß =
sin 2a + sin 2y
sin 2ß=
1+sin 2a sin 2y
11
382. If tan O= —
then sin
is
44
Lor –
a.


but not
b.
but not
d. none of these
11
39Illustration 2.5 If (sec A + tan A) (sec B + tan B) (sec C +
tan C) = (sec A – tan A) (sec B -tan B) (sec C- tan C), prove
that the value of each side is +1.
11
40In right angle ( triangle A B C, angle B=90^{circ}, angle A= )
( 45^{circ}=angle C, ) then ( operatorname{cosec} 45^{circ}= )
( mathbf{A} cdot mathbf{1} )
B. ( sqrt{2} )
c. ( frac{1}{sqrt{2}} )
D. None of above
11
41( (sin theta-sec theta)^{2}+(cos theta-operatorname{cosec} theta)^{2}= )
( (1-sec theta cdot operatorname{cosec} theta)^{2} )
11
42Solve :
( frac{1+cos x}{sin x cos x} )
11
435. If in a SABC a cos” () +ccos? ()
5.
If in a AAB
COS
, then the sides
then the sides
[2003]
a, bandc
(a) satisfy a +b=C
(b) are in A.P
(c) are in G.P
(d) are in H.P.
11
44The equation ( sin ^{2} theta-frac{4}{sin ^{3} theta-1}=1 )
( frac{4}{sin ^{3} theta-1} ) has
A. No root
B. One root
c. ( T w o ) roots
D. Infinite roots
11
45Illustration 3.32 Prove that cos a + cos ß + cos y + cos(a +
a + B β +γ γ +α
B+ ) = 4 cos —
2
2
:
COS
COS
11
4686. In any triangle ABC, sin A – sinB + sinʼC is always equal
to
a. 2 sin A sin B cos C b. 2 sin A cos B sin C
c. 2 sin A cos B cos C d. 2 sin A sin B sin C
11
47[
mathrm{f}_{mathrm{f}} prod_{r=4}^{8} cos left(frac{theta}{2^{r}}right)=frac{sin left(frac{theta}{2^{n_{1}}}right)}{(2)^{n_{2}} sin left(frac{theta}{2^{n_{3}}}right)}
]
then the value of ( n_{1}+n_{3}-n_{2} ) is
11
48and
Illustration 3.82 In triangle ABC, if cot A cot C =
cot B. cot C= =, then the value of tan C is
N-
d
11
49If ( cos 2 B=frac{cos (A+C)}{cos (A-C)}, ) then
( tan A, tan B, tan C ) are in
A. ( A . P . )
в. ( G . P )
c. ( H . P )
D. None of these
11
50The general solution of ( 4 tan ^{2} theta= ) ( mathbf{3} sec ^{2} boldsymbol{theta} ) is ( boldsymbol{theta}=boldsymbol{n} boldsymbol{pi} pm frac{boldsymbol{pi}}{boldsymbol{m}} . ) Then, find the
value of ( boldsymbol{m} )
11
5110. If 2 sec A – secA – 2 cosec-A + cosec- A = 15/4, then
tan A is equal to
a. 1/2
b. 1/2
c. 1/2 √2
d. -1/2
11
52If ( frac{3 pi}{4}<alpha<pi, ) then ( sqrt{2 cot alpha+frac{1}{sin ^{2} alpha}} )
is equal to
A. ( 1-cot alpha )
B. ( 1+cot alpha )
c. ( -1+cot alpha )
D. ( -1-cot alpha )
11
53( frac{tan ^{2} theta}{1+sec theta}+1 ) equals to
( A cdot tan theta )
B. ( frac{1}{cos theta} )
( mathbf{c} cdot sec theta-1 )
( mathbf{D} cdot sec theta+tan theta )
11
54Prove that:
( cot ^{2} frac{pi}{6}+operatorname{cosec} frac{5 pi}{6}+3 tan ^{2} frac{pi}{6}=6 )
11
55Convert ( pi / 6 ) rad to degrees.11
56Express the following angles into radian
( 10^{circ}, 40^{circ}, 30^{circ} )
11
57Find the least value of ( 2 sin ^{2} theta+3 cos ^{2} theta )11
58Illustration 3.58 If cos 0 = cos a cos B, prove that
ata O-a
tan
– = tan
– tan –
11
5912. The maximum value of cos?(45° + x) + (sin x – cos x)2 is11
60The value of ( cos ^{2} A+cos ^{2} B- )
( mathbf{2} cos boldsymbol{A} cos boldsymbol{B} cos (boldsymbol{A}+boldsymbol{B})- )
( sin ^{2}(boldsymbol{A}+boldsymbol{B}) )
11
61Find the radian measure of the interior
angle of regular hexagon.
11
62If ( boldsymbol{pi}=mathbf{1 8 0}^{circ} ) and ( boldsymbol{A}=frac{boldsymbol{pi}}{boldsymbol{6}}, ) prove that
( frac{(1-cos A)(1+cos A)}{(1-sin A)(1+sin A)}=frac{1}{3} )
11
63( cos ^{2} alpha-sin ^{2} alpha=tan ^{2} beta . ) then show
that ( tan ^{2} alpha=cos ^{2} beta-sin ^{2} beta )
11
64Prove that: ( tan left(11 frac{1^{circ}}{4}right)= )
( sqrt{4+2 sqrt{2}}-(sqrt{2}+1) )
11
65In a right triangle ( Delta A B C ), right angled
at ( B . ) If ( tan A=1, ) then verify that
( 2 sin A cdot cos A=1 )
11
6611. If p cosec 0+q cot 0 = 2 and p2 cosec? 0 – q? cot 0 = 5
then the value of 181p-2-9-2 is
11
67For ( -frac{pi}{2}<theta<frac{pi}{2}, ) range of ( f(theta)= )
( frac{sin theta+sin 2 theta}{1+cos theta+cos 2 theta} ) is
A. ( (-infty, infty) )
a
в. (-2,2)
( c cdot(0, infty) )
D. none of these
11
68Find the value of other five
trigonometric ratios:
( tan x=-frac{5}{12}, x ) lies in second quadrant.
11
69If ( alpha ) and ( beta ) are angles in the first quadrant ( tan alpha=frac{1}{7}, sin beta=frac{1}{sqrt{10}} ) then
using the formula ( sin (A+B)= )
( sin A cos B+cos A sin B ) one can find
the value of ( (boldsymbol{alpha}+mathbf{2} boldsymbol{beta}) ) to be
A ( cdot 0^{circ} )
B . ( 45^{circ} )
( c cdot 60 )
D. ( 90^{circ} )
11
70If ( sin ^{2} theta=frac{1}{4}, ) then the general solution
of ( boldsymbol{theta} )
A ( cdot 2 n pi pm(-1)^{n} frac{pi}{6} )
в. ( frac{n pi}{2} pm(-1)^{n} frac{pi}{6} )
c. ( n pi pm frac{pi}{6} )
D. ( 2 n pi pm frac{pi}{6} )
11
71Illustration 3.33 Prove that
sin A + sin 2A+ sin 4A + sin 5A
-= tan 3A.
cos A + cos 2A + cos 4A + cos 5A
11
72If ( sin theta=frac{1}{2}, cos phi=1, ) where ( 0<theta<frac{pi}{2} )
and ( 0<phi leq frac{pi}{2}, ) then ( (cot (theta+2 phi))^{2} ) is
equal to:
11
73X
3. cose cos/0 – 27 cose + 1
9.
If
then x + y + z is
cosa
2 л
<r), the
COS
O *
equal to
a. 1
b. O
d. none of these
c. – 1
O
11
74Find the value of ( sin 22^{0} cos 38^{0}+ )
( cos 22^{0} sin 38^{0} )
( A cdot frac{1}{2} )
B. ( -frac{sqrt{3}}{2} )
c. ( frac{sqrt{3}}{2} )
D. ( frac{1}{sqrt{2}} )
11
75A solution of the equation ( 5 sin ^{2} x+ )
( 3 sin x cos x-3 cos ^{2} x=2 ) is
This question has multiple correct options
A ( cdot_{2 pi+tan ^{-1} frac{-3+sqrt{69}}{6}} )
B. ( quad 7 pi+tan ^{-1} frac{-3-sqrt{69}}{6} )
c. ( tan ^{-1} frac{-3+sqrt{69}}{6}-pi )
D. ( tan ^{-1} frac{-3-sqrt{69}}{6}-5 pi )
11
7600
5.
For all ein [0, r/ 2] show that, cos (sin o) 2 sin (cos O).
(1981 – 4 Marks)
11
77If ( sin alpha sin beta-cos alpha cos beta+1=0 ) then
show that ( 1+cot alpha tan beta=0 )
11
7831. Letf(n) = 2 cos nx ne N, then f (1) f(n + 1)-f(n) is
equal to
Bilma. f(n+3)
b.f(n+2) – O na
c. f(n + 1)f (2) d. f(n + 2)f (2) leto
11
79if ( sin theta=-frac{4}{5}, pi<theta<frac{3 pi}{2}, ) then find
1. ( sin 2 theta )
2. ( cos 2 theta )
3. ( tan 2 theta )
11
80The number of value of ( x ) in ( [0,2 pi] )
satisfying the equation
( |cos x-sin x| geq sqrt{2}, ) is
11
81Solve ( 6 sin ^{2} x+2 sin ^{2} 2 x=5 )11
82(i) If ( sin (theta+alpha)=cos (theta+alpha) ) then
express tan ( theta ) in terms of ( alpha )
(ii) Find the value of ( tan (pi / 4+ )
( theta) cdot tan (pi / 4-theta) )
11
83sin4t + cost-1
4. The value of 3 –
is equal to
sint + cosºt – 1
11
84Illustration 3.78
If A + B + C = n, prove that

sin
-sin
= 1 – 2 cos
COS
11
85Solve: ( cos ^{7}left(sin frac{4 pi}{3}right) )11
86Illustration 2.50 Find the value of x for which f(x) =
sin x – cos x is defined, x € [0,21].
11
876. Let n be a positive integer such that sin
+COS
2n
=v”. Then
b. 4 <n<
to gali
a. 6Sn38
c. 45n38
d. 4<n<8 (IIT-JEE 1994)
11
8871. Which of the following is not the solution of the equation
sin 5x = 16 sinºx (n e Z)?
a. nt
.
b. nt +
c. nt –
d. none of these
11
89What is the value of ( left(frac{1}{sin 45^{0}}-sin 45^{0}right)left(frac{1}{cos 45^{0}}-cos 45^{0}right. )11
90sin4 x – cos4 x+sin? x cos? x
97. If y=
sin* x + cos* x + sin? scosx*e 0:2), then
a. susc. lsys
c.
VI
ys1
d. none of these
11
91The value of ( 36^{circ} ) in radians is
A ( cdot frac{pi}{2} )
в. ( frac{2 pi}{5} )
c.
D. ( 3 pi )
11
92Define radian measure of an angle.11
93Prove that ( : frac{sin A}{cot A+operatorname{cossec} A}=2+ )
( frac{sin A}{operatorname{Cot} A-operatorname{cosec} A} )
11
9494. Ifu=va? cos? + b2 sine + Va? sin? 6 + b2 cos? o,
then the difference between the maximum and minimum
values of u- is given by
a. 2(a² +6²) b. 2 a² +6² .
c. (a + b)2
d. (a – b)?
11
95Solve ( 1+2 operatorname{cosec} x=-frac{sec ^{2}(x / 2)}{2} )11
96If ( [sin x]+[sqrt{2} cos x]=-3, x in[0,2 pi] )
([.] denotes the greatest integer function), then ( x ) belongs to
A ( cdotleft(pi, frac{5 pi}{4}right) )
B . ( left[pi, frac{5 pi}{4}right] )
c. ( left(frac{5 pi}{4}, 2 piright) )
D・ ( left[frac{5 pi}{4}, 2 piright] )
11
9715. The respective values of tan A, tan B and tan Care
a. 1, ſ3, ſ b. 1, ſ3, 2
1. c. 1, 2, 13 d. 1, 13, 2 + V3
11
989. Prove that 1 + cotes cot – for 0 < €< it. Find when
equality sign holds.
11
99The value of ( frac{1-tan ^{2} 15}{1+tan ^{2} 15} ) is
( A cdot 1 )
B. ( sqrt{3} )
( c cdot frac{sqrt{3}}{2} )
D.
11
10046. (1 + tan a tan B)2 + (tan a – tan B)2 =
a. tan’a tan
B b . sec?a secaß
c. tan’a cot+ß iced d. sec’a cos?B
11
1016. Without using tables, prove that (sin 12°) (sin 48°)
(sin 549) = 1/8.
(IIT-JEE 1980)
11
102If ( 3 Theta ) is an acute angle, solve the
following equation ( Theta ) :
( (operatorname{cosec} 3 Theta-2)(cot 2 Theta-1)=0 )
11
103If ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma}, boldsymbol{delta} ) are the smallest ( + ) ive angles
in ascending order of magnitude which have their sines equal to a +ive quantity ( lambda ) then the value of ( 4 sin frac{alpha}{2}+ ) ( 3 sin frac{beta}{2}+2 sin frac{gamma}{2}+sin frac{delta}{2}= )
A ( cdot 2 sqrt{1-lambda} )
B. ( 2 sqrt{1+lambda} )
c. ( 2 sqrt{lambda} )
D. ( 2 sqrt{lambda+2} )
11
1049. sin (90° + 0) is
(a) sin o
(c) -cos e
(b) cos 0
(d) – sin e
11
105Illustration 2.62 Find the sign of the values of tan 113° –
cos 107° = a and tan 107° – cos 105° = b.
11
106Solve ( sin ^{4} x+cos ^{4} x=5 / 8 )11
1071. Find the coordinates of the points of intersection of the
curves y = cosx , y= sin 3x if- sus
11
108Assertion
( (A): frac{tan 3 x-tan 2 x}{1+tan 3 x tan 2 x}=1 Rightarrow x= )
( n pi+frac{pi}{4}, n in l )
Reason
( (R): tan x ) is not defined at ( x=n pi+ )
( frac{pi}{2}, n in l )
A. Both (A) and (R) are individually true and (R) is the correct explanation of (A)
B. Both (A) and (R) are individually true but (R) is not the correct explanation of ( (A) )
C. (A) is true but (R) is false
D. (A) is false but (R) is true
11
109Find the radian measures
corresponding to the following degree measures:(i) ( 25^{circ}left(text { ii) }-47^{circ} 30^{prime}left(text { iii) } 240^{circ}right.right. )
( (i v) 520^{circ} )
11
110Prove that
[
frac{sin mathbf{A}+sin 3 A+sin 5 A+sin 7 A}{cos A+cos 3 A+cos 5 A+cos 7 A}=
]
( tan 4 A )
11
111The value of ( cos left[frac{1}{2} cos ^{-1}left(cos left(frac{-14 pi}{5}right)right)right] ) is/are:
( ^{mathbf{A}} cdot cos left(frac{-7 pi}{5}right) )
B・ ( sin left(frac{pi}{10}right) )
c. ( cos left(frac{2 pi}{5}right) )
D. ( -cos left(frac{3 pi}{5}right) )
11
112For ( left(boldsymbol{theta}_{1}, boldsymbol{theta}_{2}, boldsymbol{theta}_{3}, ldots ldots boldsymbol{theta}_{n}right) boldsymbol{epsilon}(mathbf{0}, boldsymbol{pi} / 2) ) if
( ln left(sec theta_{1}-tan theta_{1}right)+ln left(sec theta_{2}-right. )
( left.tan theta_{2}right)+ldots . ln left(sec theta-tan theta_{n}right)+ln pi= )
0 then find the value of ( cos left[left(sec theta_{1}+right.right. )
( left.tan theta_{1}right)left(sec theta_{2}+tan theta_{2}right) dots . .left(sec theta_{n}+right. )
( left.tan theta_{n}right) )
11
113If the radian measures of two angles of a triangle are as given below. Find the radian measure and the degree measure of the third angle.
(i) ( frac{5 pi}{9}, frac{5 pi}{18} )
(ii) ( frac{3 pi}{5}, frac{4 pi}{15} )
11
11414. Prove that 2 cos 2″ @ +1
— = (2cos 6 – 1)(2cos 20 – 1)
2 cos 0+1
2012
(cos 22 0 – 1) … (2cos 2n-1 0-1).
11
115Determine range ( boldsymbol{y}=mathbf{3} sin boldsymbol{x}+ )
( 4 cos (x+pi / 3)+7 )
A ( cdot 7-sqrt{5-3 sqrt{3}}, 7+sqrt{5-3 sqrt{3}} )
B・ ( 7-sqrt{10-3 sqrt{3}}, 7+sqrt{10-3 sqrt{3}} )
c. ( _{7}-sqrt{10+3 sqrt{3}}, 7+sqrt{10-3 sqrt{3}} )
D ( cdot 7-sqrt{10+3 sqrt{3}}, 7-sqrt{10-3 sqrt{3}} )
11
116=an
tany
, (
By) then sin 2a+ sin2B+
tan ß
tan (a + ß – y)
tan (a – B + 7)
sin2y=
a. 0
c. 2
b. 1 o lepszywe
d. 1/2
11
117Illustration 3.72 Prove that sin 0 + sin 30 + sin 50 + …
sinʼne
+ sin(2n-1) 0=
sin e
11
118find the value
( tan 11 frac{pi}{3} )
11
119( sin 3 x+sin x+2 cos x=sin 2 x+ )
( 2 cos ^{2} x )
11
12071. iftam B-2 sin a sin ycose( a + y, then coto, coth, coty
71. If tan ß= 2 sin a sin y cosec(a + ), then cot a, cot B, coty
o are in
a. A.P.
b. G.P.
c. H.P.
d. none of these
11
121The measure of an angle in degrees, grades and radians be ( mathrm{D}, mathrm{G} ) and ( mathrm{C} )
respectively, then relation between them ( frac{boldsymbol{D}}{mathbf{9 0}}=frac{boldsymbol{G}}{mathbf{1 0 0}}=frac{boldsymbol{2 C}}{boldsymbol{pi}} ) but ( mathbf{1}^{circ}= )
( left(frac{180}{pi}right)^{0} simeq 57^{circ}, 17^{prime}, 44.8^{prime prime} ) and sum of
interior angles of a ( n ) -sided regular polygon is ( (2 n-4) frac{pi}{2} . ) On the basis of above information, answer the following questions :The number of sides of two
regular polygon are as 5: 4 and the difference between their angles is ( frac{pi}{20} )
then the number of sides in the
polygons respectively are –
A. 25, 20
B. 20, 16
c. 15,12
D. 10,8
11
122The general value of ( x ) for the equation
( 9^{cos x}-2.3^{cos x}+1=0 )
A ( . n pi )
в. ( frac{n pi}{2} )
( c cdot 2 n pi )
D. ( (2 n+1) frac{pi}{2} )
11
123Illustration 3.35 Prove that (cos a + cos 3)2 + (sin a + sin B)?
= 4 cos? (a-B)
11
124а
76. If 0=3a and sin
=-
=, the value of the expression
x 22 the
a cosec a – b sec a is
a
b. 2ſa² +6²
c. a + b
d. none of these
11
1258. If a < 3 cos x + 5 sin(x – 1/6) < b for all x, then (a, b) is
a. (-V19, 719) b. (-17, 17)
c. (-√21, √21) d. none of these
11
12629. The system of equations tan x = a cot x, tan 2x = b cos y
a. cannot have a solution if a=0
b. cannot have a solution if a=1
c. cannot have a solution if 2Va > b(1 – a)
d. has a solution for all a and b
11
127Change the following radian measure to degree measure:
( frac{3 pi}{2} )
A ( cdot 120^{circ} )
( ^{circ} )
B ( .240^{circ} )
( c .270^{circ} )
D. ( 300^{circ} )
11
128Convert ( left(frac{5 pi}{6}right)^{c} ) into degrees.11
129Prove that:
( 2 sin ^{2} frac{pi}{6}+operatorname{cosec}^{2} frac{7 pi}{6} cos ^{2} frac{pi}{3}=frac{3}{2} )
11
130( frac{sec 8 theta-1}{sec 4 theta-1}=frac{tan r theta}{tan 2 theta} cdot ) Find ( r )11
1312 TC
210
Illustration 3.93
Prove that 4cos
.cos–1=2 cos-
11
13227. The greatest value of sin+e+ cose is
a. 1/2
b. 1
c. 2
d. 3
11
133Illustration 3.83 If cos (A + B+C)=cos A cos B cos C, then
8 sin (B + C) sin (C + A) sin (A + B)
find the value of –
sin 2 A sin 2B sin 2C
11
134Solve the following equation:
( cos x=frac{1}{2} )
11
13514. If sin 0+ cos 0 = – and 0 s < , then tan o is
a. – 4/3
b. – 3/4
c. 3/4
d. 4/3
11
13667. If o is an acute angle and
tan 0 + cot 0 = 2, then the value
of tans 0 + cot5 O is
(1) 1
(2) 2
(3) 3
(4) 4
11
137( tan ^{-1}left[frac{a cos x-b sin x}{b cos x+a sin x}right]=tan ^{-1}left(frac{a}{b}right) )
( mathcal{L} )
A. True
B. False
11
138If ( cos left(65^{0}-Aright) cos left(25^{0}+Bright)- )
( sin left(65^{0}-Aright) sin left(25^{0}+Bright)=sin (m+ )
( A-B) ).Find ( m )
11
139Given ( frac{pi}{2}<alpha<pi, ) then the expression ( sqrt{frac{1-sin alpha}{1+sin alpha}}+sqrt{frac{1+sin alpha}{1-sin alpha}}= )
A ( frac{1}{cos alpha} )
B. ( -frac{2}{cos alpha} )
c. ( frac{2}{cos alpha} )
D. None of these
11
140Express the following angles in degrees.
( begin{array}{ll}text { (1) } & left(frac{5 pi}{12}right)^{circ}end{array} )
(2) ( -left(frac{7 pi}{12}right)^{circ} )
(3) ( frac{pi}{3} )
(4) ( frac{5 pi^{circ}}{6} )
(5) ( frac{2 pi^{circ}}{9} )
(6) ( frac{7 pi^{circ}}{24} )
11
141Solve ( sin x+sqrt{3} cos x geq 1 )11
14290. If cosA + cos²B + cos²C = 1, then A ABC is
a. equilateral b . isosceles
c. right angled d. none of these
11
143Illustration 3.21 Prove that (1 + tan 1°)(1 + tan 2°) …
(1 + tan 45º = 223.
11
1448. Prove that the equation 2 sin x = (x + a has no solution for
ae
3
Donne
o
ttomu
11
145( sin 11^{circ} 19 cos 18^{circ} 41+ )
( cos 11^{circ} 19 sin 18^{circ} 41= )
A .
B. ( frac{sqrt{3}}{2} )
( c cdot frac{1}{2} )
D. 0
11
146( boldsymbol{A}+boldsymbol{B}=frac{boldsymbol{pi}}{mathbf{3}} ; cos boldsymbol{A}+cos boldsymbol{B}=mathbf{1}, ) value of
( |cos A-cos B| ) is
A ( cdot frac{1}{3} )
B. ( sqrt{frac{2}{3}} )
( c cdot sqrt{frac{3}{2}} )
D.
11
147For an acute angle, ( alpha, sin alpha+cos alpha )
takes the
greatest value when ( alpha ) is
A ( .30^{circ} )
B . ( 45^{circ} )
( c cdot 60^{0} )
D. ( 90^{circ} )
11
148( tan 100^{circ}+tan 125^{circ}+ )
( tan 100^{circ} tan 125^{circ} ) is equal to
A . 0
B. ( frac{1}{2} )
( c .-1 )
D.
11
149Illustration 3.73
Prove that
cos 3x
sin 2x sin 4x
cos 5x
sin 4x sin 6x
cos 7x
+ –
sin 6x sin 8x
+
cos 9x
sin 8x sin 10x
– (cosec x) [cosec 2x – cosec 10x]
11
150Illustration 4.3 Solve tan x + tan 2x + tan 3x = tan x tan 2x
tan 3x, xe [0, 1].
11
151If ( 0 leq x leq pi ) and ( 81^{sin ^{2} x}+81^{cos ^{2} x}=30 )
then ( x ) is equal to
This question has multiple correct options
A ( cdot frac{pi}{6} )
в.
c. ( frac{5 pi}{6} )
D. ( frac{2 pi}{3} )
11
152+
54. Tet (1-сos 2x+sin 2x (1+cotx + cot’ x
(1+cos 2x + sin 2x ) ( 1+tan x + tan- x
then the minimum value of P(x) equals sro
a. 1
b. 2
c. 4
d. 16
11
15327. If COS X _ cos(x+0)_cos(x +20)
cos(x +30)
d
then
a
b
ngh atc :
is equal to
b+d
b. –
11
1545. Let f:(-1, 1) + R be such that f(cos 40) = 5
2-sece
(
π
π
for
for 0 €(0.4)-(* 1). Then the v
e
1. Then the value(s) of
is (are)
b. 1+
c.
1-
d. 1+
11
155Which of the following is least? (All angles have been measured in radians)
( A cdot sin 3 )
B. ( sin 2 )
( c cdot sin 1 )
D. ( sin 7 )
11
156satisfying
4. Find all values of 0 in the interval
the equation
(1 – tan 0) (1 + tan ) sec+ 2 tan’e = 0.
11
157If ( theta_{1}, theta_{2}, theta_{3}, ldots . theta_{n} ) are in ( A . P ., ) then
( frac{sin theta_{1}+sin _{2}+ldots+sin theta_{n}}{cos theta_{1}+cos theta_{2}+.+cos theta_{n}}= )
( A cdot 0 )
B ( cdot tan left(theta_{1}+theta_{n}right) )
c. ( tan left(frac{theta_{1}+theta_{n}}{2}right) )
D. ( tan left(frac{theta_{n}-theta_{1}}{2}right) )
11
158If ( sin x=sin y ) and ( cos x=cos y, ) then
( x ) is
A ( .2 n pi+y )
в. ( 2 n pi-y )
c. ( n pi+y )
D. ( n pi-y )
11
159Given that ( cos 50^{circ} 20^{prime}=0.6388 ) and
( cos 50^{circ} 42^{prime}=0.6334, ) then the possible
value of ( cos 50^{circ} 20^{prime} ) is
A .0 .6293
B. 0.6307
c. 0.6361
D. 0.6414
11
160Find the general solution of the
equation ( 4 cos ^{2} x=1 )
11
161OS
у
4. If sinx + cosx = y + – for x € [0, 1], then
a. x = 7/4
b. y = 0
c. y = 1
d. x = 370/4
11
16218. If (1 + sin t) (1 + cot t) =
– then find the value of
(1-sin t) (1 – cos t).
11
16332. If 0< 0< t, then minimum value of 3 sin 0+ cosec O is
a. 4
b. 3
c. 5
d. 6
11
16488. In triangle ABC,
csin A+sin B + sin C
sin A+ sin B-sin C
is equal to
a. tan
cot”
2
2
А
В
cot cota
d. tantan
11
165Find the value of ( tan 9^{circ}-tan 27^{circ}- )
( tan 63^{circ}+tan 81^{circ} )
( mathbf{A} cdot mathbf{0} )
B . 2
c. 1
D. 4
11
16650. One of the general solutions of 4 sinºx + cos x = 1 is
a. nt = a/2, a=cos (1/5), V ne z
b. nn a/2, a= cos(3/5), V ne z
c. 2n = a/2, a= cos'(1/3), V ne z
d. none of these
11
167Solve the following equation:
( cos x=sqrt{3} )
11
168( f sin left(60^{circ}+30^{circ}right)=sin 60^{circ} cos 30^{circ}+ )
( sin 30^{circ} cos 60^{circ} ) then what is the value of
( sin left(60^{circ}+30^{circ}right) .=? )
11
169Illustration 3.8 In AABC, if cot A + cot B + cot C = 0 then
find the value of cos A cos B cos C. 20
11
170Prove that
( cos frac{pi}{15} cos frac{2 pi}{15} cos frac{3 pi}{15} cos frac{4 pi}{15} cos frac{5 pi}{15} cos frac{6}{1} )
1
( 2^{7} )
11
171Illustration 3.25
Find the range of
f(x)=
(cos x – 3)2 + (sin x +4)2
11
1721. (1+cos 1 + cos y 1 + cos 4 + cos?”) is equal
– COS
(1984 – 3 Marks)
(6) cos.
(d) 1+ v2
(d) 22
11
173The number of solutions of the equation ( 2 cos left(frac{x}{2}right)=5^{x}+5^{-x} ) is
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D. None of these.
11
1743-tan?
55. If –
= k cos – then the value of k is
1 – tan2
a. 1
c. 3
b. 2
op d. 4
11
175( cos ^{2} 42^{circ}-sin ^{2} 48^{circ} )11
176Illustration 3.91
1/16.
Prove that sin 6° sin 42° sin 66° sin 78°
11
17721. Which of the following is not the value of sin 27° –
cos 27°?
b. – V5-15
2
c.- 212
d. none of these
11
17874. The smallest positive x satisfying the equation
logcosxsin x + logsinr cos x = 2 is
a. Td/2
b. /3
c. 77/4 shot
d. Td/6
11
179Illustration 3.49 Find the maximum and minimum values
of cos 0 – 6 sin cos 0+3 sin’e+2.
11
1804. If 3 tan A + 4 = 0, then the value of 2 cot A-5 cos A+
sin A is equal to
4. If3 tan A + 4 = 0, then the value of 2 cot A -5 cos A+
a.
if –
<A<TT
<A<2a
2
<A< T
d.
<A< 211
10
10
11
18114.
The general values of
2sin20-3sino-2=0 is
satisfying the equation
(1995)
(b) na +(-1)”</2
(a) na +(-1)” x/6
©
nt+(-1)"51/6
(d) nt+(-1)" 770/6
11
182The degree measure of 1 radian (taking ( left.boldsymbol{pi}=frac{boldsymbol{2} boldsymbol{2}}{boldsymbol{7}}right) ) is
A ( cdot 55^{circ} 61^{prime} 22^{prime prime} ) (approx.)
В ( cdot 57^{circ} 16^{prime} 22^{prime prime} ) (approx.)
c. ( 57^{circ} 22^{prime} 16^{text {” }} ) (approx.)
D. ( 57^{circ} 22^{prime} 22^{prime prime} ) (approx.)
11
183If ( 2 tan beta+cot beta=tan alpha, ) prove that
( cot beta=2 tan (alpha-beta) )
11
184If ( 0<x leq frac{pi}{2}, ) then ( (sin x+operatorname{cosec} x) ) is
greater than or equal to
A . 0
B.
( c cdot 2 )
D. None of these
11
185If ( x sin ^{3} theta+y cos ^{3} theta=cos theta sin theta ) and
( boldsymbol{x} sin boldsymbol{theta}=boldsymbol{y} cos boldsymbol{theta} operatorname{then} boldsymbol{x}^{2}+boldsymbol{y}^{2}=mathbf{1} )
If the statement is True, enter 1 , else
enter 0
A.
11
186If ( sin ^{2} theta+5 cos ^{2} theta=4, ) then find ( theta ) and
hence prove that ( sec theta+operatorname{cosec} theta=2+ )
( frac{2}{sqrt{3}} )
11
18713.
tan x whe
Show that the value of a -, wherever defined never lies
tan 3x
between – and 3.
(1992 – 4 Marks)
11
188If ( tan x=-frac{3}{4}, frac{3 pi}{2}<x<2 pi, ) then find
( cos 2 x )
11
189Prove that:
( sin 50^{circ}+sin 10^{circ}=cos 20^{circ} )
11
19066. Which one of the following is true
for 0° < cos20
(3) cose cos20
11
191As ( theta ) increases from ( frac{pi}{4} ) to ( frac{5 pi}{4}, ) the value
of ( 4 cos frac{1}{2} theta )
A. increases, then decreases
B. decreases, and then increases
C. decreases throughout
D. increases throughout
E. decreases, increases, and then decreases again
11
1923. The general value of 0 satisfying the equation tan²0 +
sec 20= 1 is
(IIT-JEE 1996)
11
193cos x
18. If
sin x
siny
1
2
3
= =, where x, y e
2′
1, then the
cos y
value of tan(x + y) is equal to
a. 113
c. 17
b. 114
d. 115
11
19446._
sin? A-sin-B
is equal to
sin Acos A-sin B cos B
a. tan(A – B)
c. cot(A – B)
= b. tan(A + B)
d. cot(A + B)
11
19513. 3 (sin x -cos x)* + 6 (sin x + cos x)2 + 4 (sinºx+cos© x) =
(1995)
(a) 11 (6) 12 C 13 (d) 14
11
196ff ( boldsymbol{x}=boldsymbol{a} sin boldsymbol{theta}+boldsymbol{c} cos boldsymbol{theta} ) and ( boldsymbol{y}= )
( boldsymbol{a} cos boldsymbol{theta}-boldsymbol{c} sin boldsymbol{theta}, operatorname{then} boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{a}^{2}+boldsymbol{c}^{2} )
A . True
B. False
11
197In the fourth quadrant, and
trigonometric ratios are positive
A . ( cos ), sec
B. sin, cos
( c cdot sin , operatorname{cosec} )
D. tan, cot
11
198( frac{sin (n+1) A+2 sin n A+sin (n-1) A}{cos (n-1) A-cos (n+1) A} )
is equal to
A ( cdot tan frac{A}{2} )
B. ( cot frac{A}{2} )
( c cdot tan A )
D. ( cot A )
11
199If ( A ) and ( B ) are acute positive angles satisfying the equations ( 3 sin ^{2} A+ ) ( 2 sin ^{2} B=1 ) and ( 3 sin 2 A-2 sin 2 B= )
( 0, ) then ( A+2 B ) is equal to
A ( cdot frac{pi}{4} )
в.
( mathrm{c} cdot_{3} frac{pi}{4} )
D. ( frac{2 pi}{3} )
11
200then find the range of
Illustration 3.4 If sin a cos B=-
values of cos a sin ß.
11
201If ( x=sin 1, y=sin 2 ; z=sin 3 ) then
A. ( x<yy>z )
c. ( y<z<x )
D. ( z<x<y )
11
202The value of
( sin 10^{circ} sin 30^{circ} sin 50^{circ} sin 70^{circ} ) is:
( ^{A} cdot frac{1}{36} )
в. ( frac{1}{32} )
c. ( frac{1}{18} )
D. ( frac{1}{16} )
11
20331. Number of real solutions of the equation (tan x + 1)
(tan x + 3) (tan x + 5) (tan x + 7) = 33
a. will be two in the interval [– 1/2, 7/2]
b. will be four in the interval [- r/2, 1/2]
c. will be three in the interval (-1/2, 1)
d. will be four in the interval (-1/2, 1)
11
204show that ( sqrt{2+sqrt{2+sqrt{2+2 cos 8 theta}}}= )
( 2 cos theta, 0<theta<frac{pi}{8} )
11
205Illustration 3.83 If cos (A+B+C)=cos A cos B cos C, then
8 sin (B + C) sin (C + A) sin (A + B)
find the value of
sin 2 A sin 2B sin 2C
11
206Illustration 3.79 In any triangle ABC, prove that
sin’A cos(B – C) + sin’B cos(C – A) + sinC cos(A – B)
= 3 sin A sin B sin C
11
207State true or false.
If ( left(1+sin ^{2} thetaright)=3 sin theta cos theta ) then
( tan theta=-1 o r frac{1}{2} )
A . True
B. False
11
208Prove the following Identities ( frac{tan alpha+tan beta}{cot alpha+cot beta}=tan alpha tan beta )11
209Solve the following equation:
( sin x=frac{sqrt{2}}{2} )
11
210Illustration 4.36 Find common roots of the equations 2sin²x
+ sinº 2x = 2 and sin 2x + cos 2x = tan x.
11
2112. The value of tan ß is
sin a(1+ Acos B)
Acos a cos ß
sin a(l – Acos B)
Acos a cos ß
cosa(1+ Asin ß)
A cos a cos ß
cosa(1 – Asin B)
Acos a cos ß
11
212If ( boldsymbol{alpha} boldsymbol{epsilon}left(boldsymbol{0}, frac{boldsymbol{pi}}{2}right), ) then the expression
( sqrt{x^{2}+x}+frac{tan ^{2} x}{sqrt{x^{2}+x}} ) is always greater
than or equal to
( A cdot 2 tan alpha )
B. 2
( c . )
( D cdot sec ^{2} alpha )
11
213In which quadrant does the terminal
side of the angle ( 250^{0} ) lie?
A. Quadrant III
B. Quadrant I
c. Quadrant
D. Quadrant IV
11
214Illustration 2.44
Find the range of f(x) =
4 cos x
3
11
215What is ( (sin x cos y+ )
( cos x sin y)(sin x cos y-cos x sin y) )
equal to?
A ( cdot cos ^{2} x-cos ^{2} y )
B. ( cos ^{2} x-sin ^{2} y )
( c cdot sin ^{2} x-cos ^{2} y )
D. ( sin ^{2} x-sin ^{2} y )
11
216Let ( alpha ) and ( beta ) be any two positive values
of ( x ) for which ( 2 cos x,|cos x|, ) and ( 1- )
( 3 cos ^{2} x ) are in G.P. The minimum value
of ( |boldsymbol{alpha}-boldsymbol{beta}| ) is
A ( cdot frac{pi}{3} )
в.
c. ( frac{pi}{2} )
D. None of these
11
21724. If a = *then the value of (tan a tan 2c + tan 2a tan 4a.
14°
+ tan 4a tan a) is
a. 1
b. 1/2
c. 2
d. 1/3
11
218Find all pairs of ( x, y ) that satisfy the
equation ( cos x+cos y+cos (x+y)=-3 / 2 )
11
219The value of ( sin ^{2} 30^{circ}-cos ^{2} 30^{circ} ) is:
A. ( -frac{1}{2} )
B. ( frac{sqrt{3}}{2} )
( c cdot frac{3}{2} )
D. ( frac{2}{3} )
11
220State whether the following statement is true or false.
( frac{cos A-sin A+1}{cos A+sin A-1}=operatorname{cosec} A+cot A )
(by using the identity ( operatorname{cosec}^{2} boldsymbol{A}=1+ )
( left.cot ^{2} A .right) )
A . True
B. False
11
221Illustration 3.69 Show that
4 sin 27° = (5+15)1/2 – (3-55)12.
11
222If ( boldsymbol{x}=boldsymbol{a}+boldsymbol{b} boldsymbol{omega}+boldsymbol{c} boldsymbol{omega}^{2}, boldsymbol{y}=boldsymbol{a} boldsymbol{omega}+boldsymbol{b} boldsymbol{omega}^{2}+boldsymbol{c} )
and ( z=a omega^{2}+b+c omega ) then find the
value of ( frac{x^{2}}{y z}+frac{y^{2}}{z x}+frac{z^{2}}{x y} )
11
223If ( frac{cos (theta-alpha)}{sin (theta+alpha)}=frac{m+1}{m-1}, ) then ( m ) is equal
to
A ( cdot tan left(frac{pi}{4}-thetaright) tan left(frac{pi}{4}-alpharight) )
B. ( tan left(frac{pi}{4}-thetaright) tan left(frac{pi}{4}+alpharight) )
c. ( tan left(frac{pi}{4}+thetaright) tan left(frac{pi}{4}+alpharight) )
D. ( tan left(frac{pi}{4}+thetaright) tan left(frac{pi}{4}-alpharight) )
11
224If ( sin x+sin ^{2} x=1, ) then ( cos ^{2} x+cos ^{4} x )
is :
( A )
B. 2
( c cdot 3 )
D. 4
11
225Illustration 3.61 Evaluate cos a cos 2a cos 3a … cos 999a,
211
where a=
1999
11
226( cos ^{4} theta-sin ^{4} theta+1 ) is equal to:
A ( cdot 2 cos e c^{2} theta )
B. ( -2 cos e c^{2} theta )
c. ( frac{2}{tan ^{2} theta} )
D. ( frac{2 cot ^{2} theta}{cos e c^{2} theta} )
11
2274. In which of the following sets the inequality sinºx + cosºx
> 5/8 holds good?
a. (-1/8, 7/8)
b. (37/8, 57/8)
c. (Tt/4, 31/4)
d. (771/8, 9/8) .
11
22819. Show that
1+ sin A
cos A 1

+-
cos B
-sin B
2 sin A-2 sin B
sin(A-B)+cos A – cos B
11
2292. Which of the inollowing is collectin 1° sin 1
b. sin 1° < sin 1
c. sin 1° = sin 1
b
d. sin 1° =
*
sin 1
180
11
230All the angles between and which
satisfy ( 90^{circ} ) the ( 0^{circ} ) equation ( sec ^{2} theta cdot cos e c^{2} theta+2 cos e c^{2} theta=8 )
11
231If ( sec A=operatorname{cosec} B=frac{5}{3}, ) then the value
of ( (boldsymbol{A}+boldsymbol{B}) ) is equal to
( mathbf{A} cdot mathbf{0} )
B. ( 90^{circ} )
( mathbf{c} cdot90^{circ} )
11
23285. If x = r cos O cos 0, y = r cos
sin 0 and 2 = r sin 0, then the
value of x2 + y + z is
(1) 2
(2)
11
233equals
38. The value of expression
2(sin 1° + sin 2° + sin 3° + … + sin 89°)
2
2(cos 1° +cos 2° +…+cos 44°)+1
a. 12
b. 1712
c. 1/2
d. O
10
11
234If ( sin x-cos y=sin frac{pi}{7} ) and ( cos x+ )
( sin y=cos frac{pi}{7} ) then find the value of
( sin (x-y) )
11
235If ( sin A=frac{5}{13} ) then evaluate ( cos A ) and
( tan A )
11
236( frac{sin A}{sin left(90^{circ}-Aright)}+frac{cos A}{cos left(90^{circ}-Aright)}= )11
237Illustration 4.22 Find the general values ofx and y satisfying
the equations 5 sin x cos y = 1 and 4 tan x = tany.
11
238If ( tan (A+B)=p, tan (A-B)=q )
then prove that ( tan 2 A=frac{p+q}{1-p q} )
11
239Illustration 4.39
Solve 7 cos²0 + 3 sin²0= 4
11
24054. The value of
(+tan? 2°. tan88°
(1) 1
(2) 2
(3) O
(4) 4
11
241Show that ( frac{2 tan 30^{circ}}{1-tan ^{2} 30^{circ}}=sqrt{3} )11
24213. If O E TO, 57) and r e R such that 2sin e = r4 – 2r-
+ 3 then the maximum number of values of the pair
(r, ) is
11
243Solve ( : log _{frac{x^{2}-6 x}{10}}(sin 3 x+sin x)= )
( log _{frac{x^{2}-6 x}{10}}(sin 2 x) )
A. ( _{x=frac{5 pi}{3}} )
в. ( x=-frac{5 pi}{3} )
c. ( x=-frac{2 pi}{3} )
D. ( x=-frac{4 pi}{3} )
11
244Prove ( frac{cos 20^{circ}+sin 20^{circ}}{cos 20^{circ}-sin 20^{circ}}=tan 65^{circ} )11
245Find ( cos ^{4}(pi / 8)+cos ^{4}(3 pi / 8)+ )
( cos ^{4}(5 pi / 8)+cos ^{4}(7 pi / 8)-3 / 2 )
11
246Solve the equation ( sqrt{3} cos x-sin x=1 )
A ( cdot x=2 n pi-frac{pi}{3} pm frac{pi}{3}, n in Z )
В . ( x=2 n pi-frac{pi}{6} pm frac{pi}{3}, n in Z )
c. ( x=2 n pi-frac{pi}{6} pm frac{pi}{6}, n in Z )
D. None of these
11
247Solve: ( 2(sin x-cos 2 x)-sin 2 x(1+ )
( 2 sin x)+2 cos x=0 )
11
248solve:-
( 1+cos 2 x+cos 4 x+cos 6 x= )
A ( .2 cos x cos 2 x cos 3 x )
B. 2 ( cos x sin 2 x cos 3 x )
( mathrm{c} cdot 4 cos x cos 2 x cos 3 x )
D. ( 4 cos x sin 2 x sin 3 x )
11
249Which of the following is correct?
( A cdot sin 1^{circ}>sin 1 )
B. ( sin 1^{circ}<sin 1 )
( mathbf{c} cdot sin 1^{circ}=sin 1 )
D ( cdot sin 1^{circ}=frac{pi}{180} sin 1 )
11
2505. The number of solutions of the equation tan x + sec x =
2 cos x lying in the interval [0, 21) is
a. O
b. 1
c. 2
d. 3 (IIT-JEE 1993)
11
2514. Given a + B – y = 1, prove that sin’a + sin?B – sin?y=
2 sin a sin ß cos y.
(IIT-JEE 1980)
11
2526.
Without using tables, prove that
(1982 – 2 Marks)
(sin 12°) (sin 48) (sin
11
25336. The general solution of tan 0 + tan 20 + tan 30= 0 is
a. 0=nt/6, ne Z only
b. O=nt &, ne Z, where tan a= 1/
c. Both a and b
d. none of these
11
254Convert ( 290^{circ} ) into radian measure11
255Illustration 2.29 Suppose the point with coordinates (-12,5)
is on the terminal side of angle 8. Find the values of the six
trigonometric functions of e.
11
256( operatorname{can} tan 65^{0}=tan 25^{0}+tan 40^{0} ? )
If Yes answer is ( 1, ) else 0
11
257Illustration 3.95
Illustration 3.95
* 5 = 1, then find the range of 2x + y.
=1, then find the range of 2x + y.
11
258Express the following angles in radian
measure:
i) ( 520^{circ} )
ii) ( -310^{circ} )
iii) ( 630^{circ} )
iv) ( -22^{circ} 30^{prime} )
11
259If ( sin A=frac{1}{2} ) and ( cos B=frac{sqrt{3}}{2} ) where ( A )
lies in second quadrant and B lies in first quadrant, find the values of (a) ( sin (A-B)(b) cos (A-B)(c) tan (A+ )
( boldsymbol{B}) )
11
26050. If tan Atan B = =, then (5 – 3 cos 2A) (5 – 3 cos 2B) =
a. 2
b. 8
d. 16
11
261If ( tan ^{2} ) la ( tan ^{2} beta+tan ^{2} beta cdot tan ^{2} gamma+ )
( tan ^{2} gamma cdot tan ^{2} alpha+2 tan ^{2} alpha cdot tan ^{2} beta cdot tan ^{2} gamma= )
1, then the value of ( sin ^{2} alpha+sin ^{2} beta+ )
( sin ^{2} gamma ) is :
A .
B. –
( c )
( D cdot frac{1}{2} )
11
2621.
The period of sine is
(a) a? (b) a
©
20
[2002]
(d) a 12
11
263Given ( sin phi=frac{15}{17}, ) find the value of ( frac{3-4 sin ^{2} phi}{4 cos ^{2} phi-3} )11
264If ( 2 sin left(theta+frac{pi}{3}right)=cos left(theta-frac{pi}{6}right), ) prove
that ( tan theta+sqrt{3}=0 )
11
265Illustration 4.45
Solve 13 cos 0 – 3 sin 0 = 4 sin 20 cos 30.
11
266In the figure given above, ( A B ) is paralle
to ( C D . ) If ( angle D C E=x ) and ( angle A B E=y )
then what is ( angle C E B ) equal to?
A ( cdot y-x )
в. ( frac{(x+y)}{2} )
c. ( x+y-(pi / 2) )
D. ( x+y-pi )
11
26741. If cos? x -(c – 1) cos x + 2c 26 for every x e R, then the
true set of values of c is
a. [2,-)
D. 14, ) GHT 02
c. (-∞, -21
d. (-0, 4]
11
26817. If и, = sin”Ө + cos”Ө, thеn рrоvе thаt 5
и; – и, — и
1 = 3.
и — и, и
11
26979. If V2 cosA=cosB + cos’ B, and V2 sin A=sin B-sin’ B
then sin (A – B) =
a. +1
bゃう
c. +
atd.
d.t-
11
270Simplify: ( sin ^{-1}(cos x) )11
271( operatorname{In} ) any triangle ( prodleft(frac{sin ^{2} B+sin B+1}{sin B}right) )
is always greater than
( A cdot 9 )
B. 3
c. 27
D. None of these
11
272Prove ( : sqrt{frac{1+sin A}{1-sin A}}=sec A+tan A )11
273If ( tan theta+frac{1}{tan theta}=2, ) find the value of
( cot ^{2} theta+frac{1}{cot ^{2} theta} )
11
2745. Let 0 < x < 7:/4, then (sec 2x -tan 2x) equals
X
in
x
+
11
275( sin ^{-1} frac{2 a}{1+a^{2}}+cos ^{-1} frac{1-b^{2}}{1+b^{2}}= )
( 2 tan ^{-1} x )
11
276Prove that ( frac{cos 9^{0}+sin 9^{0}}{cos 9^{0}-sin 9^{0}}=cot 36^{0} )11
277alue for which tan o– 1, cos o-ta
2. The most general value for which tan O=-1, cos O=
is (n e Z)
a. nt+
b. nt + (-1)” ITT
c. 2n1 + 711
d. none of these
11
27814. The value of tan A tan B + tan B tan C + tan C tan A IS
a. 5 – 4 13 b. 5 +4 13
c. 6 + √3
d. 6-√3
11
27910
2. Compute tan 22
11
2802. Given to the
3.
Given a+B-y=t, prove that
sina + sin B-sin?y=2 sina sin
prove singsing cosy
cosy
(1980)
1980)
11
281tan 20
15. Prove that an 2 = (1 + sec 20) (1 + sec 220)
tan e
(1 + sec 230) … (1 + sec 2″0).
11
282If ( 0 leq x leq 2 pi, ) then the number of
solutions of the equation ( sin ^{6} x+ )
( cos ^{6} x=1 ) is
( A cdot 2 )
B. 3
( c cdot 4 )
D. 5
E . 8
11
283Three roots of the equation, ( x^{4}-p x^{3}+ )
( boldsymbol{q} boldsymbol{x}^{2}-boldsymbol{r} boldsymbol{x}+boldsymbol{s}=boldsymbol{0} ) are ( tan boldsymbol{A}, tan boldsymbol{B} boldsymbol{&} )
( tan C ) where ( A, B, C ) are the angles of ( a ) triangle. The fourth root of the bi
quadratic is
A ( frac{s^{2}-s q+s}{r+(s-q) p} )
B. ( frac{s^{2}+s q+s}{r+(s+p) q} )
c. ( frac{s^{2}-s q+s}{r-(s-p) q} )
D. ( frac{s^{2}-s q-s}{r-(s-q) p} )
11
284Sin
56. The total number of solutions of cotx| = cotx +
e [0, 31), is equal to
a. 1
b. 2
c. 3
d. 0
11
285If lies in the second quadrant, then the value of ( sqrt{left(frac{1-sin theta}{1+sin theta}right)}+sqrt{left(frac{1+sin theta}{1-sin theta}right)} ) is
( mathbf{A} cdot 2 sec theta )
в. ( -2 sec theta )
( c .2 operatorname{cosec} theta )
D. noneofthese
11
28613. One root of the equation cox – * * } – oles in the
73. One root of the equation cos x – x + – = 0 lies in the
interval
2
T,
11
287m=0
1. Suppose sin’x sin 3x = Cocos mx is an identity in x,
where Co, C, … ,C, are constants, and Cn #0, then the
value of n is
(IIT-JEE 1981)
11
2887. Solve the equation 2 sinx + cos y = 2 for the values of x
and y.
11
289Is it right to say that ( sin (A+B)= )
( sin A+sin A . ) Justify your answer?
11
29025. If x and y are positive acute angles such that (x + y) and
(x – y) satisfy the equation tan-0 – 4 tan 0+1= 0, then
b. y =
d. x
11
291If ( cos x=tan y, cot y=tan z ) and
( cot z=tan x ; ) then ( sin x= )
A. ( frac{sqrt{5}+1}{4} )
B. ( frac{sqrt{5}-1}{4} )
c. ( frac{sqrt{5}+1}{2} )
D. ( frac{sqrt{5}-1}{2} )
11
292If ( sin x+sin y=sqrt{3}(cos y-cos x) )
then value of ( sin 3 x+sin 3 y ) equals
( A cdot 3 )
B. ( sqrt{3} )
( c cdot 0 )
D.
11
293Prove that ( tan (x-y)=frac{tan x-tan y}{1+tan x tan y} )11
294ff ( boldsymbol{y}=sec ^{2} boldsymbol{Theta}+cos ^{2} boldsymbol{Theta}, boldsymbol{0} neq boldsymbol{0}, ) then
A. ( y=0 )
B. ( y leq 0 )
( mathbf{c} cdot y>2 )
D. ( y neq 2 )
11
295The range of ( f(x)=cos ^{2} x+sec ^{2} x ) is
( [a, infty] ) Find a
11
296If ( tan A-tan B=x ) and ( cot B- )
( cot A=y ) then ( cot (A-B)= )
A ( cdot frac{1}{y}-frac{1}{x} )
в. ( frac{1}{x}-frac{1}{y} )
c. ( frac{1}{x}+frac{1}{y} )
D. None of these
11
297%. Given both o and are noue avec amd sin = 12,
9. Given both O and o are acute angles and sin 0 = 1/2,
cos 0 = 1/3, then the value of 8 + o belongs to
17
.
a 2n
a.
57
3
11
2988. For the equation 1- 2x – x2 = tan?(x + y) + cot(x + y)
a. exactly one value of x exists
b. exactly two values of x exists
c. y=-1 + nnt + 7/4, ne z
d. y= 1 + nn + /4, ne Z
11
299A wheel makes 240 revolutions in one
minute The measure of the angle it
describes at the centre in 15 seconds is
( mathbf{A} cdot 60 pi )
в. ( 120 pi )
( c cdot 8 pi )
D. ( pi )
11
300Write the following relation in the Roster
form and hence find its domain and
range ( R=left{left(a, a^{2}right) / a ) is a prime right.
number less than ( 15} )
11
301If ( alpha+beta=frac{pi}{2} ) and ( sin alpha=frac{1}{3}, ) then ( sin beta )
is equal to
A. ( frac{sqrt{2}}{3} )
B. ( frac{2 sqrt{2}}{3} )
( c cdot frac{2}{3} )
D. 3
11
3022. Which of the following statements are always correct
(where Q denotes the set of rationals)?
a. cos 20 € Q and sin 20 € Q tano e Q (if defined)
a b. tan de Q sin 20, cos 20 and tan 20 € Q (if defined)
c. if sin 0 e Q and cos O e Q tan 30 € Q (if defined)
d. if sin 0 e Q cos 30 € Q
11
303Illustration 3.87 Prove that
sin 10° sin 30° sin 50° sin 70º = 1/16.
11
304The solution of the equation ( 9 cos ^{12} x+ )
( cos ^{2} 2 x+1=6 cos ^{6} x cos 2 x+ )
( 6 cos ^{6} x-2 cos 2 x ) is/are:
( mathbf{A} cdot mathbf{x}=mathbf{n} pi+frac{pi}{2} ; mathbf{n} in mathbf{I} )
в. ( _{x}=n pi+cos ^{-1}(sqrt[4]{frac{2}{3}}), n in I )
c. ( _{x=n pi-cos ^{-1}}(sqrt[4]{frac{2}{3}}), n in I )
D. none of the above
11
30528. The equation 2sin? 0+ (21 – 3)sin? -(32+2)sin 0-22
= 0 has exactly three roots in (0, 21), then a can be equal
to
a. 0
b. 2
c. 1
d. -1
11
30674. tan• ** – 33 tan* * +27 tan? ” is equal to
b. 13
a. 0
c. 3
.00
d. 9
11
307Prove that ( frac{tan theta}{1-cot theta}+frac{cot theta}{1-tan theta}=1+ )
( sec theta operatorname{cosec} theta )
11
30815. If logiosin x + log1ocos x = -1 and log10(sin x + cos x) =
(log10 n)-1
-, then the value of ‘n/3′ is
2
11
309Prove that:
( cos 2 alpha cos 2 beta+sin ^{2}(alpha-beta)-sin ^{2}(alpha+ )
( boldsymbol{beta})=cos 2(boldsymbol{alpha}+boldsymbol{beta}) )
11
31056. sin 105° + cos 105° will be
equal to
(1) sin 45º (2) tan 45º
(3) cosec 45° (4) sec 45°
11
3119. ABC is a triangle such that sin(2A + B) = sin(C – A) =
– sin(B + 2C) = 1/2. If A, B and C are in A.P. determine
the values of A, B, and C.
(IIT-JEE 1990)
11
312Find the value of ( sin left(15^{0}right) )
A ( frac{sqrt{3}-1}{2 sqrt{2}} )
B. ( frac{sqrt{3}+1}{2 sqrt{2}} )
c. ( frac{sqrt{3}-1}{sqrt{2}} )
D. None of these
11
313If ( A ) is in the 3 rd quadrant and ( B ) is in the
fourth quadrant and ( cos A= ) ( -frac{15}{17}, cos B=frac{4}{5} ) find the value of
( cos (boldsymbol{A}+boldsymbol{B}) )
A ( cdot-frac{84}{85} )
в. ( frac{84}{85} )
( c cdot-frac{34}{85} )
D. None of these
11
314If ( boldsymbol{x}=boldsymbol{y} cos frac{boldsymbol{2} boldsymbol{pi}}{mathbf{3}}=boldsymbol{z} cos frac{boldsymbol{4} boldsymbol{pi}}{boldsymbol{3}}, ) then ( boldsymbol{x} boldsymbol{y}+ )
( y z+z x ) is equal to
( mathbf{A} cdot-4 x^{2} )
B.
( c cdot 1 )
D.
11
315( frac{1+sin 2 theta}{1-sin 2 theta}=left(frac{1+tan theta}{1-tan theta}right)^{b} )
Find ( b )
11
31634. The total number of solution of sin^x + cos4x = sin x cos x
in [0, 2n) is equal to
a. 2
b. 4
c. 6
d. none of these
11
3174. Let 2 sinºx + 3 sin x – 2 > 0 and x2 – x – 2 < 0 (x is
measured in radians). Then x lies in the interval
51108 028
b.
c. (-1,2)
IT-JEE 1994)
11
318The value of ( frac{1}{operatorname{cosec}^{2} theta}+frac{1}{sec ^{2} theta} ) is
( mathbf{A} cdot mathbf{1} )
B.
( c cdot sin ^{2} theta )
( mathbf{D} cdot cos ^{2} theta )
11
31989. Find the number of pairs of integer (x, y) that satisfy the
following two equations:
cos(xy) = x
tan (xy) = y
b.
2a
od d. 6
0
a. 1
c. 4
11
32063. The simplest value of cot 9° cot
27° cot 63° cot 81° is
(1) 0
(2) 1
(3) -1 (4) 3
11
3211. The value of the expression
tan²20° – sin²20°
tan²20°sin20°
is
11
322Evaluate:
( frac{5 sin ^{2} 30^{0}+cos ^{2} 45^{0}+4 tan ^{2} 60^{0}}{2 sin 30^{0} cos 60^{0}+tan 45^{0}}=frac{55}{m} )
Find ( m )
11
32364. If cos 40º = a, then value of
cos 100° will be
(1) 1-2a2 (2) 2a2 – 1
(3) 2a +1 (4) 2a
11
324Let ( A B C ) be a acute angled triangle such that ( A=frac{pi}{3} ) and ( cos B cos C=P )
The possible range of values of ( boldsymbol{P} ) will be
A ( cdotleft(-frac{1}{4}, frac{1}{4}right] )
в. (0,1]
( c cdotleft[frac{1}{3}, inftyright) )
( mathbf{D} cdot[1, infty) )
11
3251. Which of the following number(s) is/are rational?
are a. sin 15°
b. cos 15°
c. sin 15º cos 15° d. sin 15° cos 75°
11
326Simplify ( 2 x^{2}+y^{2}+2 x y=5 ) where,
( x=(2 cos theta-sin theta) ) and ( y=(cos theta) )
( 3 sin theta) )
11
32734. If x, and x2 are two distinct roots of the equation a cos x
+ b sinx = c, then tan ***2 is equal to
a.
b. 6
11
328The number of values of ( x ) in the interval
( [0,5 pi] ) satisfying the equation ( 3 sin ^{2} x-7 sin x+2=0 ) is
A .
B. 5
( c cdot 6 )
D. 10
11
329Obtain the value of ( frac{cos 45^{circ}}{sec 30^{circ}+operatorname{cosec} 30^{circ}} )11
3303. The number of distinct solutions of the equation
cos? 2x + cos+ x + sin* x + cos x + sinº x = 2 in the
interval [0, 21) is
(JEE Advanced 2015)
11
33115. If (cosec?0 – 4)x2 + (cot 8 + V3 ) x + cos2 34 = 0 holds
true for all real x, then the most general values of e can
be given by (ne Z)
a. 2n1 + 111
b. 2nT +
6
c. 2nt +
d. nt
11
332In which quadrant is ( theta ), if ( sin theta ) is
positive and ( cos theta ) is negative?
( A )
B. I
( c )
D. IV
11
333The value of the expression ( frac{2left(sin 1^{circ}+sin 2^{circ}+sin 3^{circ}+ldots+sin 89^{circ}right)}{2left(cos 1^{circ}+cos 2^{circ}+ldots+cos 44^{circ}right)+1} )
is
A ( cdot sqrt{2} )
в. ( frac{1}{sqrt{2}} )
( c cdot frac{1}{2} )
( D )
11
334Illustration 4.24
Solve sin 20+ cos 0 = 0.
11
33511. If cot + tan 0 = x and sec -cos O = y, then
a. x sin 8. cos 0 = 1 b. sin?o= y cos e
c. (x+y)1/3 + (xy2)1/3 = 1 d. (x+y)2/3 – (xy2)2/3 = 1
11
33640. Number of solutions of the equation cos+ 2x + 2sin? 2x =
17(cos x + sin x)8, 0<x<2n is
b. 8
c. 10
d. 16
a. 4
11
337Prove that:
( frac{cos theta}{1-sin theta}+frac{cos theta}{1+sin theta}=2 sec theta )
11
33871. If sin 0 – cos 0 = 1 and
0< < 90°, then the value of sin
0 + cos 0 is
(3 ja
47
11
339When ( n ) is an odd natural number other
than ( 1, ) then the value of ( x ) is
This question has multiple correct options
A. ( -pi / 2 )
B.
( c . pi )
D. ( 3 pi )
11
340If ( A B=-bar{i}-2 bar{j}-6 bar{k}, B bar{C} C=2 bar{i}-bar{j}+ )
( bar{k}, bar{A} C=bar{i}-3 bar{j}-5 bar{k} . ) Then ( angle B=? )
A ( cdot )
[
begin{array}{l}cos ^{-1}(sqrt{frac{40}{41}}) \ text { B } cdot \ qquad begin{array}{l}text { cos }^{1}(sqrt{frac{6}{41}}) \ text { c. } cos ^{-1}left(frac{6}{41}right)end{array} \ text { D. } cos ^{-1}left(frac{62}{63}right)end{array}
]
11
341Sol. We have n sin x = mcos xl
Draw the graphs of y=nsin x and y=mcos xl.
Ranges of n sin x and micos x are fo, n] and [0, m],
respectively.
Also, period of each of nisin x and micos x is T.
Graphs of functions are as shown in the following figure.
y = nisin x1
Xv = m/cos x B x
to
save

X
Td2
372
27
5:22
Fig. 4.4
From the figure graphs intersect at four points.
Hence, there are four roots of the equation.
For point A, n sin x = m cos x
tan x =
r=tan-1 m
For point B, x = – tan-
For point C, x = 11 + tan-1
For point D, x = 21 – tan!
11
342If ( A+B=225^{circ}, ) then
( frac{cot boldsymbol{A}}{mathbf{1}+cot boldsymbol{A}} cdot frac{cot boldsymbol{B}}{mathbf{1}+cot boldsymbol{B}}= )
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 0 )
D. ( frac{1}{2} )
11
343Illustration 3.50 Show that V2 + 12 + 12 + 2 cos 80 =
2 cos 0,0<</16.
11
344Evaluate ( sin 29^{circ}-cos 61^{circ} )11
345<a<, then
a. 1 + cota
c. 1- cota
2 cota + – is equal to
sina
b. -1 – cota
d. -1 + cota
11
346Find the number of solutions of the
equations ( (sin x-1)^{3}+(cos x-1)^{3}+(sin x)^{3}= )
( (2 sin x+cos x-2)^{3} ) in ( [0,2 pi] )
11
3475. The value of tan a tan ſ tan y tand is
a. – 1/3
b. -2
c. 0
d. none of these
11
348( cos ^{2} frac{3 pi}{5}+cos ^{2} frac{4 pi}{5} ) is equal to
A ( cdot 4 / 5 )
B. ( 5 / 2 )
c. ( 5 / 4 )
D. ( 3 / 4 )
11
34922)
41. The number of values of 0 in the interval
IT I
satisfying the equation (v3)** ° = tan^ +2 tan’e is
b. 4
200
c. 0
d. 1
a. 2
11
350The value of ( tan 75^{circ} ) is
A ( cdot 1+frac{1}{sqrt{3}} )
B. ( 2-sqrt{3} )
( c cdot 2+sqrt{3} )
D. ( 1+sqrt{3} )
11
351The solution of ( (2 cos x-1)(3+ )
( 2 cos x)=0 ) in the interval ( 0 leq theta leq 2 pi )
is
( mathbf{A} cdot(pi / 3) )
В. ( (pi / 3,5 pi / 3) )
c. ( [pi / 3,5 pi / 3] )
D. none of these
11
352The value of ( cot 16^{circ} cot 44^{circ}+ )
( cot 44^{circ} cot 76^{circ}-cot 76^{circ} cot 16^{circ} ) is
A . 3
B. ( frac{1}{3} )
( c cdot-frac{1}{3} )
D. – 3
11
35352. The value of cos y cos
COS X +
52. The value of cosy cos (5. – 2) – cos (7. – y) cosx +
siny cos ( 6 – x) + cosx sin (7. – ») is zero if
a. X=0
b. y = 0
c. x = y
d. nt + y –
(ne Z)
11
354Assertion
In a triangle ( A B C ) if ( tan A: tan B: )
( tan C=1: 2: 3 ) then ( A=45^{circ} )
Reason
If ( p: q: r=1: 2: 3 ) then ( p=1 )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
11
355Illustration 4.34
Solve tan 50= cot20.
11
35666. The value of sec2 12°–
tan2 78°
(1) 0
(2) 1
(4) 3
(3) 2
11
357+
Illustration 3.73 Prove that
cos 3x cos 5x
cos 7x
cos 9x
sin 2x sin 4x sin 4x sin 6x sin 6x sin 8x’ sin 8x sin 10x
(cosec x) [cosec 2x – cosec 10x]
11
358Prove that:
( 2 sin ^{2} frac{3 pi}{4}+2 cos ^{2} frac{pi}{4}+2 sec ^{2} frac{pi}{3}=10 )
11
35915. Solve the equation for x, sin10x + cosi’x = 29 cos4 2x.
16
11
36035. General solution of tan 0 + tan 40 + tan 70 = tan tan 40
tan 70 is
a. O=nt/12, where ne Z
b. 0=nt/9, where ne Z
c. 0= nt + Tt/12, where ne Z
d. none of these
11
3618. The value of cos(a + B) is
12
b. ?
25
13
d. none of these
11
362If OCR
and cos X
sin
then tan x is
() -(4+57)
d (1+17)
11
363Illustration 3.86 Prove that
cos 20° cos 40° cos 60° cos 80º = 1/16.
11
36486. The total number of ordered pairs (x, y) satisfying [x] + byl
= 2, sin (ntx{/3) = 1, is equal to
a. 2
b. 3
d. 6
c. 4
11
365( operatorname{Prove} sin left(mathbf{6 0}^{0}+boldsymbol{theta}right)-sin left(mathbf{6 0}^{mathbf{0}}-boldsymbol{theta}right)= )
( sin boldsymbol{theta} )
11
366Illustration 2.6 Given that sin 30º = 1/2 and cos 30º =
V3/2. Determine the values of sin 60°, sin 120°, sin 240°,
sin 300°, and sin (-30%).
11
367( frac{cos 15^{circ}-sin 15^{circ}}{cos 15^{circ}+sin 15^{circ}}=frac{1}{sqrt{3}} )11
368Prove that
[
begin{array}{l}
(cos alpha+cos beta)^{2}+(sin alpha+sin beta)^{2}= \
4 cos ^{2}left(frac{alpha-beta}{2}right)
end{array}
]
11
369Find the degree measure of the angle
subtended at the centre of a circle
of radius ( 100 mathrm{cm} ) by an arc of length 22
( mathrm{cm} .left(boldsymbol{U s e} boldsymbol{pi}=frac{mathbf{2 2}}{mathbf{7}}right) )
11
370If ( 1+sin x+sin ^{2} x+sin ^{3} x+ldots infty ) is
equal to ( 4+2 sqrt{3}, 0 leq x<pi ) then ( x ) is
equal to
A.
в.
c. ( frac{pi}{3} ) or ( frac{pi}{6} )
D. ( frac{pi}{3} ) or ( frac{2 pi}{3} )
11
37111. Number of solutions of the equation sinºx – cos²x sinx
+ 2 sin x + sinx = 0 in 0 SXS 31 is
11
372Illustration 2.55 If 0 < a < p < y< 1/2, then prove that
sin a + sin ß + sin y
tan a<
cosa + cos ß + cos y
– <tan y.
11
373The set of angles between ( 0 & 2 pi )
satisfying the equation ( 4 cos ^{2} theta ) ( 2 sqrt{2} cos theta-1=0 ) is –
A ( cdotleft{frac{pi}{12}, frac{5 pi}{12}, frac{19 pi}{12}, frac{23 pi}{12}right} )
В ( cdotleft{frac{pi}{12}, frac{7 pi}{12}, frac{17 pi}{12}, frac{23 pi}{12}right} )
c. ( left{frac{5 pi}{12}, frac{13 pi}{12}, frac{19 pi}{12}right} )
D. ( left{frac{pi}{12}, frac{7 pi}{12}, frac{19 pi}{12}, frac{23 pi}{12}right} )
11
37426. If x, y, z are in A.P, then
sin x -sin z
is equal to
a. tany
c. siny
COS Z – cos x
b. coty
d. cos y
11
375Illustration 2.42 If sin?, + sin²0, + sin’ex = 0, then which
of the following is not the possible value of cose + cos O2
+ cos Oz?
a. 3
b. -3
c. -1
d. -2
11
37665. If cosx = 2-cos y
2 cos y-1
where x, y E (0, ), then tan – cot
is equal to
b. 3
11
377Illustration 4.40
Solve 2 sin’x + sin2x = 2.
11
378If ( tan alpha=frac{1}{7}, tan beta=frac{1}{sqrt{10}}, ) prove that
( alpha+2 beta=frac{pi}{a}, ) where ( 0<alpha<frac{pi}{2} ) and
( mathbf{0}<boldsymbol{beta}<frac{boldsymbol{pi}}{mathbf{2}} )
Find ( a )
11
379If ( sin Theta+cos Theta=sqrt{2}, ) and ( Theta ) is actual
then ( tan Theta ) is equal to
A ( cdot frac{1}{sqrt{3}} )
B.
( c cdot sqrt{3} )
D. ( infty )
11
380the fundamental period, if any, of the
functions:
( sin (x / 3) ) is ( k pi, ) then ( k ) is
11
3817.
The general solution of
sin x-3 sin 2x + sin 3x = cos x – 3 cos 2x + cos 3x is
(1989-2 Marks)
(a) n+
(d)
2ntt + cos!
11
382Illustration 4.49 If x, y e [0, 271), then find the total
number of ordered pairs (x, y) satisfying the equation
sinx cos y = 1.
11
38369. If roots of the equation 2×2 – 4x + 2 sin 0 – 1 = 0 are of
opposite sign, then e belongs to
BULEVY
(1 51
a. (76
137
177
c. (1991)
d. (0,7)
d. (0, 1)
66
11
384( frac{tan 5 Theta+tan 3 Theta}{tan 5 Theta-tan 3 Theta}=4 cos 2 Theta cos 4 Theta )11
385( cos 1^{circ} cos 2^{circ} cos 3^{circ} dots dots dots dots dots cos 90^{circ} )
( mathbf{A} cdot mathbf{0} )
B . -1
( mathbf{c} cdot 1 )
D. None of these
11
386prove ( rightarrow frac{1+cos Theta+sin Theta}{1+cos Theta-sin Theta}=frac{operatorname{coz} Theta}{1-sin Theta} )11
38747. Ifcos(a-B)=3 sin(a+B), then –
1-3 sin 2a
1-3 sin 2B
Tail
d.
11
388Prove the following statements:
( cos ^{4} boldsymbol{A}-sin ^{4} boldsymbol{A}+mathbf{1}=boldsymbol{2} cos ^{2} boldsymbol{A} )
11
389The value of ( sin ^{2} 20^{circ}+sin ^{2} 70^{circ} ) is :
( mathbf{A} cdot mathbf{1} )
B. 0
( c cdot 2 )
D. 3
11
39012. The number of solutions of 12 cos’x – 7 cos²x + 4 cosx =
9 is
a. 0
b. 2
c. infinite
d. none of these
11
3915. Let f(x) = x² – 2/(sin 13 — sin V2) x – (cos 13 – cos 12)
then
a. f(x) is positive VxER
b. f(x) assumes both positive and negative values
c. f(x) = 0 has no real roots
d. y=f(x) touches the line y = 0.
11
392Illustration 3.34 Prove that
cos A + cos B ( sin A +sin B
sin A -sin B (cos A – cos B
B
A
in
= 2 cot”
OF O,
2
accordingly as n is even or odd.
11
39395. If tan x = n tan y. ne Rt then the maximum value of
sec-(x – y) is equal to
95. yn
myne P, then the maximum value of
b
8
a. (n+1)2
(n+1)
2
2n
b. (n+1)
c. non
(n+1)
tie 2
d. (n+12
4n
11
394Simplify ( frac{5 cos ^{2} 60^{0}+4 sec ^{2} 30^{0}-tan ^{2} 45^{0}}{sin ^{2} 30^{0}+cos ^{2} 30^{0}} )
( A cdot-frac{55}{2} )
в. ( frac{67}{12} )
c. ( frac{5}{12} )
D. ( frac{55}{4} )
11
395( sin ^{-1} x+sin ^{-1}left(frac{1}{x}right)+cos ^{-1} x+ )
( cos ^{-1}left(frac{1}{x}right)= )
( mathbf{A} cdot pi )
( B cdot frac{pi}{2} )
( c cdot frac{3 pi}{2} )
D. ( 2 pi )
11
396Illustration 2.17 By geometrical interpretation, prove that
tan a + tan B
tan(a+B) = 1
1- tan a tan ß
11
397The inequality ( 2^{sin theta}+2^{cos theta} geq 2^{1-(1 / sqrt{2})} )
holds for all real values of ( theta )
A . True
B. False
11
39884. If A + B + C = 37/2, then cos 2A +cos 2B + cos2C is equal
to
a. 1 – 4 cos A cos B cos C
b. 4 sin A sin B sin C
c. 1 + 2 cos A cos B cos C
d. 1 -4 sin A sin B sin C
11
399Illustration 4.32
Solve tan 30=-1.
11
400Evaluate each of the following:
( frac{sin 60^{circ}}{cos ^{2} 45^{circ}}-cot 30^{circ}+15 cos 90^{circ} )
11
40161. The value of cot 70° + 4 cos 70° is
b. √3
c. 2/3
d1 0800
2
11
402The number of roots of the equation ( x+2 tan x=frac{pi}{2} ) in the interval ( [0,2 pi] ) is
( mathbf{A} cdot mathbf{1} )
B . 2
( c cdot 3 )
D. Infinite
11
403Solve for ( x ) and ( y: ) ( boldsymbol{x}+boldsymbol{y}=frac{boldsymbol{7} boldsymbol{pi}}{boldsymbol{4}}, frac{sin boldsymbol{x}}{cos boldsymbol{y}}=-sqrt{boldsymbol{2}} )11
404Find the value of:
( 2 tan 45^{circ}+cos 45^{circ}-sin 45^{circ} )
11
405( fleft(x-frac{x^{2}}{4}+frac{x^{3}}{4}-dots dots inftyright)+ )
( cos ^{-1}left(x^{2}-frac{x^{4}}{4}+frac{x^{6}}{4}-ldots ldots . . inftyright)=frac{pi}{2} )
and ( 0<x<sqrt{2} ) then ( x= )
A ( cdot frac{1}{2} )
B.
( c cdot-frac{1}{2} )
D. –
11
4069.
For a positive integer n, let
(1999 – 3 Marks)
CO
1+ sec ) (1+sec 20) (1 + sec 40)….(1 + sec 2″).
Then
o sa (5)=1
@ ss (153)=1
11
407Illustration 3.11 Let a, Band y satisfy 0<a<B Y 2 .1
cos (x + a) + cos (x + B) + cos (x + y) = ( for all x ER, then
find the possible values of (Y-a).
11
408( frac{sin (B-C)}{cos B cos C}+frac{sin (C-A)}{cos C cos A}+ )
( frac{sin (A-B)}{cos A cos B}= )
11
409Convert ( 25^{circ} ) into radian.11
410Illustration 4.60 Prove that the least positive value of x,
satisfying tan x = x + 1, lies in the interval (71/4, 1/2).
11
411( operatorname{Prove} cos ^{2} alpha-sin ^{2} 2 alpha= )
( cos ^{2} alpha cos 2 alpha-2 sin ^{2} alpha cos ^{2} alpha )
11
412The measure of an angle in degrees, grades and radians be ( mathrm{D}, mathrm{G} ) and ( mathrm{C} ) respectively, then relation between them ( frac{D}{90}=frac{G}{100}=frac{2 C}{pi} ) but ( 1^{circ}= )
( left(frac{180}{pi}right)^{circ} simeq 57^{circ}, 17^{prime}, 44.8^{prime prime} ) and sum of
interior angles of a ( n ) -sided regular polygon is ( (2 n-4) frac{pi}{2} . ) On the basis of above information, answer the following questions :Which of the following are correct
This question has multiple correct options
( A cdot sin 1^{circ}cos 1^{circ} )
( mathbf{c} cdot cos 1^{circ}<cos 1^{c} )
D. ( 1^{c}=57^{circ} Rightarrow sin 1^{c}cos 57 )
11
413The value of ( 9 tan ^{2} theta-9 sec ^{2} theta ) is
( mathbf{A} cdot mathbf{1} )
B. 0
( c .9 )
D. – –
11
414Which quadrant ( 180<theta<270 )
degrees lies?
A . quadrant I
B. quadrant III
c. quadrant!
D. quadrant IV
11
415If ( cos left(frac{3 pi}{4}+xright)-cos left(frac{3 pi}{4}-xright)= )
( -sqrt{m} sin x . ) Find ( m )
11
41612. The expression cos(a + b + cos-(a – b) – cos 20.
cos 2ß, is
a. independent of a
b. independent of ß
c. independent of a and ß
d. dependent on a and ß.
11
417If ( x=cos 10^{circ} cos 20^{circ} cos 40^{circ}, ) then the
value of ( x ) is
A ( cdot frac{1}{4} tan 10^{circ} )
B. ( frac{1}{8} cot 10^{circ} )
c. ( frac{1}{8} cos 10^{circ} )
D. ( frac{1}{8} sec 10^{circ} )
11
418air of equations
4. The number of solutions of the pair of eau
2sin? 0 – cos2 0 = 0 and 2 cos? 0 – 3sin 0 = 0) in
interval [0, 21) is
a. O
b. 1
c. 2
d. 4
11
41977. The equation sinºx – 2cos x + a2 = 0 can be solved if
a. – √3 sas ſ3 b. -√2 sas √2
c. -1 Sas1
d. None of these
11
420Express the following angle into radians
( mathbf{5 0}^{circ} mathbf{3 7}^{prime} mathbf{3 0}^{prime prime} )
11
421If ( tan theta=frac{b}{a}, ) then the value of
( a cos 2 theta+b sin 2 theta ) is
( mathbf{A} cdot b )
в.
c. ( frac{a}{b} )
D. ( frac{a}{a+b} )
11
422Illustration 3.90
Prove that
147
2n 40
8T
COSCOS – COS – COS
15
15
15
15
16
11
423The value of ( frac{7 pi^{circ}}{9} ) in sexagesimal measure is
A ( .120^{circ} )
B. ( 130^{circ} )
c. ( 140^{circ} )
D. ( 150^{circ} )
11
424If ( A=60^{circ} ) verify the following.
( sin 2 A=2 sin A . quad cos A )
11
42587. The value of
3 IN
cos
is equal to
r=0
a. 1/4
c. -1/4
b. 1/8
d. -1/8
11
4267.
The number of values of x in the interval [0,3Tt] satisfying
the equation 2 sin2 x+5 sin x-3=0 is
[2006]
(2) 4 (6)
6 C 1 (d) 2
11
4275. If sin 0 – cos 0 = 1, then the value of sine – cos O is11
428The value of ( tan 105^{circ} ) is
A. ( -(2+sqrt{3}) )
B. ( -2+sqrt{3} )
c. ( sqrt{3}+sqrt{2} )
D. ( -(sqrt{3}+sqrt{2}) )
11
429Find the value of ( operatorname{cosec}^{2} 60^{circ}-tan ^{2} 30^{circ} )11
430tan? T
tan 2
.
*+tan2 31
11. The value of —
cot??
& cot2 20
7
+ cot 30
a. 7
c. 21/5
b. 35/3
d. none of these
11
43116. The value of
sin 1° + sin 3° + sin 5° + sin 7°
cos 1°•cos 2°•sin 4°
11
432Find the degree measure of ( frac{pi}{8} )11
4331. Solve 3 tan 2x – 4 tan 3x = tan²3x tan 2x.11
434The value of ( sin 50^{circ}-sin 70^{circ}+sin 10^{circ} )
is
A . 0
B.
( c cdot frac{1}{2} )
D. ( frac{1}{sqrt{2}} )
11
4354 ST
Illustration 3.54
Prove that cos*
+ cos
+ cost da
+ cos4
NIw
8
11
436tan
2
58. What will be the value of
tan 6-a) from the follow-
ing?
(1)
(1) 1+ sin A
COSA 1-sin A
(2) cos A
(3) Both of above
(4) None of the above
11
437Illustration 4.48 For what value of k the equation sin x +
cos(k +x) + cos(k – x) = 2 has real solutions?
11
4381. Find the value of sin (105°).
10
11
439Find the general solution of each of the following equations. ( sin left(x+frac{pi}{5}right)=0 )11
440If ( cos ^{2} x+cos ^{2} 2 x+cos ^{2} 3 x=1, ) then
A ( x=(2 n+1) frac{pi}{4}, n in I )
в. ( _{x}=(2 n+1) frac{pi}{2}, n in I )
c. ( _{x=n pi pm} frac{pi}{6}, n in I )
D. None of these
11
441Convert 4 radians into degree measure
and also convert ( -47^{circ} 30^{prime} ) into radian
measure.
11
44218. If cos 30= os 3 a, then the value of sin
can be given by
(T
a.
sin a
b. sin
– ta
3
(
3
11
443Solve: ( frac{cos ^{2} 25^{circ}+cos ^{2} 65^{circ}}{sin ^{2} 59^{circ}+sin ^{2} 31^{circ}}=? )
( mathbf{A} cdot mathbf{0} )
B.
c. 2
( D )
11
444Find radian measure corresponding to
the degree measure ( -37^{circ} 30^{prime} )
11
4451-tan-
Illustration 3.43
Prove that
-= sin 2 A.
1+tan?
– A)
11
446If ( operatorname{cosec} theta-cot theta=p, ) then ( operatorname{cosec} theta+ )
( cot theta= )
( A cdot 1 / p )
B. -1/p
( c cdot-p )
D ( cdot p^{2} )
11
4471. If cos(A – B) = 3/5 and tan A tan B = 2, then
a. cos A cos B =1/5 b. sin A sin B = -2/5
c. cos A cos B=-1/5 d. sin A sin B =-1/5
11
448If ( f(x)=left[cos x cos (x+2)-cos ^{2}(x+right. )
1) ( ] ) where ( [.] ) denotes the greatest integer function ( leq x ). Then solution of
the equation ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x} ) is :
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 0 )
D. none of these
11
44953. The value of sinº 10° + sinº 50°– sinº 70° is equal to
b. A
4
11
45014. Number of triangles ABC if tan A = x, tan B = x + 1, and
tan C= 1- x is
11
451UCH
74.
If sec x + cos x = 2, then the
value of sec16 x + cos 16 x will be
(1) 13 (2) 2
(3) 1
(4) O
11
452The measure of an angle in degrees,
grades and radians be ( mathrm{D}, mathrm{G} ) and ( mathrm{C} )
respectively, then relation between them ( frac{boldsymbol{D}}{mathbf{9 0}}=frac{boldsymbol{G}}{mathbf{1 0 0}}=frac{boldsymbol{2} boldsymbol{C}}{boldsymbol{pi}} ) but ( mathbf{1}^{circ}= )
( left(frac{180}{pi}right)^{0} simeq 57^{circ}, 17^{prime}, 44.8^{prime prime} ) and sum of
interior angles of a ( n ) -sided regular polygon is ( (2 n-4) frac{pi}{2} . ) On the basis of above information, answer the following questions:One angle of a triangle is ( frac{4 x}{3} )
grades and another is ( 3 x ) degrees, while the third is ( frac{2 pi x}{75} ) radians. Then the angles in degrees are –
A ( cdot 20^{circ}, 60^{circ}, 100^{circ} )
B . 24 ‘ , 60 ( ^{circ}, 96^{circ} )
c. ( 36^{circ}, 60^{circ}, 84^{circ} )
D. ( 20^{circ}, 40^{circ}, 120^{circ} )
11
453If ( sqrt{frac{1-sin A}{1+sin A}}+frac{sin A}{cos A}=frac{1}{cos A}, ) for all
permissible values of ( A, ) then ( A ) belongs
to
This question has multiple correct options
A. First Quadrant
B. Second Quadrant
c. Third Quadrant
D. Fourth Quadrant
11
45417. The value of x in (0, TT/2) satisfying –
3-
13+1
sin x cos x
= 4-2 is
11
455Illustration 4.46 Find the number of integral values of n so
that sinx (sinx + cos x) = n has at least one solution.
11
45616. If sin (sin x + cos x) = cos (cos x – sin x), and largest
possible value of sin x is
, then the value of k is
11
457Sector area of a circle in radians is
A ( cdot 2 times frac{theta}{2} r^{2} )
в. ( frac{theta}{2} times r^{2} )
c. ( _{pi times frac{theta}{2} r^{2}} )
D. ( _{2 times frac{theta}{2} pi r} )
11
458If the angles a, b, y of a triangle satisfy the relation,
2222”
16. The measure of the smallest angle of the triangle is
a. 30°
b. 40°
c. 450
d. 50°
11
45911. If sin(x + 20°) = 2 sinx cos 40º, where x € (0, 1/2), then
which of the following hold(s) good?
a. cos 2x = 1/2 b. cosec 4x = 2
c. sec * = V6 – v2 d. tan* = (2-13)
11
460Greatest possible difference between two of the roots if ( boldsymbol{theta} boldsymbol{epsilon}[mathbf{0}, boldsymbol{2} boldsymbol{pi}] ) is
A . 2
B.
( c cdot sqrt{2} )
D. ( 2 sqrt{2} )
11
46125. Let 0 € 0, and t; = (tane)tano, tz = (tane)coto,
tz =(coto) tane and t4 =(coto) cote, then (2006 – 3M, -1)
(a) 4 >t2>tz > 14
(b) to> tz>t>tz
c) tz >t>t2>14 (d) t2>tz >t>t4
3
1
11
46219. x = Va’ cosa + b2 sin’ a + Va? sina + b2 cos? a
then x2 = a + b2 +2 Vp (a? +b?) – p, where p is equal
to
a. dcos?a + b² sinʼa
af sin’a + b2 cosa
[a? + b2 + (a? – b?) cos 2a]
[a? + b2 – (a? – 62) cos 2a]
11
46355. If tanda = 1 + 2tanB, then
(90°-2a)] will be equal
(1) cos B
(2) 1 + 2cos
(3) 1 + cos28
(4) 2 cosa
11
46454. If tan 0-3sino = 0 then what
will be the value of sin’e –
coso –
(1) (1672 +1)
(2) 27 (16/2 –1)
(3) (1672 – 1)
(4) 2 (1672 +1)
11
465Find the value of ( sin 60^{circ} ) geometrically.11
466The value of ( frac{1}{cos 290^{circ}}+frac{1}{sqrt{3} sin 250^{circ}} ) is?
( A cdot frac{2 sqrt{3}}{3} )
B. ( frac{4 sqrt{3}}{3} )
( c cdot sqrt{3} )
D. None
11
467Find the degree measure of ( frac{1^{c}}{4} )11
46811. For a triangle ABC it is given that cos A+ cos B+cos C =
Prove that the triangle is equilateral.
11
469Prove the following identities. ( frac{1+sin alpha}{1+cos alpha} cdot frac{1+sec alpha}{1+operatorname{cosec} alpha}=tan alpha )11
470The value of ( cos (A+B) ) if ( sin A=frac{3}{5} )
and ( cos B=frac{8}{17} )
11
4718. The number of integral values of k for which the equation
7 cos x + 5 sin x = 2k + 1 has a solution is
a. 4
b. 8
c. 10
d. 12 (IIT-JEE 2002)
11
472The complete set of values of ( x )
satisfying equation ( cot x-cos x= )
( 1-cot x cos x ) is
A ( cdotleft{x: x=(4 n pi+1) frac{pi}{4}, n in Iright} )
в. ( left{x: x=2 n pi+frac{pi}{4}, n in Iright} )
C ( cdot{x: x=2 n pi pm pi, n in I} )
D ( cdot{x: x=2 n pi+pi, n in I} cupleft{x: x=n pi+frac{pi}{4}, n in Iright} )
11
473( boldsymbol{y}=sec ^{-1}left(frac{1}{2 x^{2}-1}right) )
Prove that: ( boldsymbol{y}=2 cos ^{-1} boldsymbol{x} )
11
4745. Solve the equation tan^x + tan^y + 2 cot?x cotły =
3+ sin? (x + y) for the values of x and y.
11
475Illustration 2.33 If sinf a + cos4 B + 2 = 4 sin a cos B,
osa, ßs then find the value of (sin a + cos 3).
11
476If ( tan A, tan B ) are the roots of ( x^{2}- )
( mathbf{2} boldsymbol{x}+mathbf{2}=mathbf{0} ) then ( cot ^{2}(boldsymbol{A}+boldsymbol{B})= )
A ( cdot frac{4}{5} )
в. ( frac{1}{2} )
( c cdot frac{3}{4} )
D.
11
477s of the equation
18. If sum of all the solutions of the
8c08* (cos+x.cos(x) 7.)-1 in (0, 1) iska
COS
[JEE M 2018)
then k is equal to :
11
478If ( tan theta+tan phi=a, cot theta+cot phi= )
( boldsymbol{b}, boldsymbol{theta}-boldsymbol{phi}=boldsymbol{alpha} neq mathbf{0}, ) then
This question has multiple correct options
( mathbf{A} cdot a b>4 )
в. ( a b<4 )
( ^{mathbf{c}} tan ^{2} alpha=frac{a b(a b-4)}{(a+b)^{2}} )
D ( sec ^{2} alpha=frac{(a-b)^{2}+a^{2} b^{2}}{(a+b)^{2}} )
11
479A ladder makes an angle ( 30^{circ} ) with the
floor and its lower end is 12 m away
from the wall. Find the length of the ladder
11
480( frac{sin 30^{circ}}{sin 45^{circ}}+frac{tan 45^{circ}}{sec 60^{circ}}-frac{sin 60^{circ}}{cot 45^{circ}}-frac{cos 30^{circ}}{sin 90^{circ}} )11
48144. If
<a<3
then
1- cos a
V1+cos a
1+cos a
– is equal to
11- cos a
2
sin a
00
a.
b
.

sin a
1
c.
sin a
R
sae d. —
o llosin a
lot
11
48214. Ifp = sin (A – B) sin (C-D), q = sin (B-C) sin (A-D),
r=sin (C – A) sin (B-D) then
a. p +q-r=0 b. p + q + r = 0
c. p-+r=0
d. p3 +93 +p3 = 3pqr
11
483If ( A+B+C=180^{circ} ) then ( a sin (B-C)+b sin (C- )
( A)+c sin (A-B) ) is :
( A cdot O )
B. 1
( c cdot 2 )
D. 3
11
484Evaluate ( cos (A+B) cdot cos (A-B)= )
( A cdot sin ^{2} A-cos ^{2} B )
B. ( cos ^{2} A+sin ^{2} B )
( mathbf{c} cdot cos ^{2} A-sin ^{2} B )
( mathbf{D} cdot cos ^{2} A+cos ^{2} B )
11
48527. The value of f(x) = x4 + 4×3 + 2×2 – 4x + 7, when x =

110
cot –
E is
11
486If ( boldsymbol{A}+boldsymbol{B}+boldsymbol{C}=boldsymbol{pi} ) and ( mathbf{A}, mathbf{B}, mathbf{C} ) are all
greater then 0 and angle ( C ) is obtuse then
( mathbf{A} cdot tan A tan B geq 1 )
B. ( tan A tan B<1 )
( mathbf{c} cdot tan A tan B=1 )
D. None of these
11
48723.
cos(a-B)= 1 and cos(a+B) = 1le where a Ber-.
Pairs of a, ß which satisfy both the equations is are
(20055
(a) 0 (b) 1 (c) 2 0 @ 4
11
488Illustration 3.101 Find the least value of sec A + sec B +
sec C in an acute angled triangle.
11
489If ( A+B=45^{circ}, ) prove that ( (cot A- )
1) ( (cot B-1)=2 )
11
490llustration 3.53
lustration 3:53 Prove that tan mo +2 tan +4 = com o
Prove that tan
Kloo
16
11
491Prove that ( : frac{1+cos A+sin A}{1+cos A-sin A}= )
( frac{1+sin A}{cos A} )
11
49240. If 2 sec 20 = tan o+cot o, then one of the values of 0+ 0
is
a. 1/2
c. 7/3
b. a /4
d. none of these
11
493Let a,ß be such that i<a-B<37.
If sin a + sin B =-
and cos a + cos B = —
then the
65
value of cosa,
[2004]
(6) Viso
11
49434. The set of values of le R such that sin? 0+ cos 0= 2 cosa e
holds for some 8, is
a. (-∞, 1]
b. (-0, -1]
c. 0
d. [-1,-)
11
495If ( sin theta=frac{45}{53}, ) find the value of
( operatorname{cosec}^{2} theta-cot ^{2} theta )
11
496What is the value of ( frac{sin ^{2} 30^{0}}{cos ^{2} 30^{0}}+ )
( frac{cos ^{2} 30^{0}}{sin ^{2} 30^{0}} ? )
11
49730. The minimum value of a tan²x + b cot?x equals the
maximum value of a sin’e + b cos?where a > b > 0.
Then alb is
a. 2
b. 4
c. 6
d. 8
11
498Illustration 2.37 Which of the following is the greatest?
a. tan 1
b. tan 4
c. tan 7
d. tan 10
11
499Illustration 3.44 Prove that (cos A – cos B)2 + (sin A – sin B)?
= 4 sin’[(A – B)/2].
11
500sin 2x
26. The least positive solution of cot
lies in
11
501If ( 3 x=sec theta ) and ( frac{3}{x}=t a n theta, ) then find the value of ( 9left(x^{2}-frac{1}{x^{2}}right) )11
502If ( alpha, beta ) are two different values of ( theta ) lying
between 0 and ( 2 pi ) which satisfy the
equation ( 6 cos theta+8 sin theta=9 ) Find
( cos (alpha+beta) )
11
503Solve for ( x, sin ^{2} 2 x=(sin 2 x) )11
5043. 3 (sin e-cos )4 + 6 (sin 0 + cos 02 +4 (sin e + cos®) is
equal to
a. 11
c. 13
b. 12
d. 14
(IIT-JEE 1995)
11
505Illustration 4.23 Solve V3 sec 20= 2.11
506For each natural number ( k ), let ( C_{k} )
denote the circle with radius ( k )
centimeters and centre at the origin ( O )
on the circle ( C_{k} ) a particle moves ( k ) centimetres in the counter-clockwise
direction. After completing its motion
on ( C_{k}, ) the particle moves to ( C_{k+1} ) in the radial direction (away from centre). The motion of the particle continues in this manner. The particle starts at ( (1,0) . ) If the particle crosses the positive direction of ( x ) -axis for the first time on
the circle ( C_{n}, ) then ( n= )
A. 4
B. 5
c. 6
D.
11
507cos(A+C)
69. If cos 2 B=
cos(A-C)
2, then tan A, tan B, tan C are in
a. A.P.
b. G.P.
c. H.P.
d. none of these
11
508Illustration 3.55
If it < x < 21, prove that
V1 + cos x + VI – cos x
V1 + cos x – V1 – cos x
== cot
+
Rt
11
509If in a ( Delta A B C, cos A . cos B+ )
( sin A cdot sin B cdot sin C=1, ) then triangle
( A B C ) is
A. isosceles
B. right angled
c. equilateral
D. right angle isosceles
11
510In a ( triangle A B C, ) the angles ( A ) and ( B ) are two
different values of ( theta ) satisfying
( sqrt{mathbf{3}} cos theta+sin theta=k,|k|<2 . ) The
triangle:
A. is an acute angled
B. is aright angled
c. is an obtuse angled
D. has one angle ( =frac{pi}{3} )
11
51126. If cotA cot?B = 3, then the value of (2 – cos 2A) (2 –
cos 2B) is
11
51229. If 2 sin? ((1/2) cos2x) = 1 – cos(a sin 2x),x + (2n+1) 7/2,
ne I, then cos 2x is equal to
a. 1/5
b. 3/5
c. 4/5
d. 1
11
51351. For n e Z, the general solution of (V3 – 1) sin 0 + ( 13
+ 1) cos 0 = 2 is (ne 2)
a. 0 = 2n1 = ” +
b. 0= nt+ (-1)”
c. 6=27=
c. O= 2nt
+
BIBIT
1
d.=na+(1)I
11
514Illustration 2.4 If 3 sin 0 + 5 cos 0 = 5, then show that
5 sin – 3 cos O=+3.
11
515If ( cos x=sqrt{1-sin 2 x}, 0 leq x leq pi, ) then
possible value of ( x ) is
This question has multiple correct options
A . ( pi )
B.
( c cdot tan ^{-1} 2 )
D. ( 3 pi )
11
516What is the simplified value of ( frac{sin 2 A}{1+cos 2 A} ? )
( mathbf{A} cdot tan A )
B. ( cot A )
( c cdot sin A )
D. ( cos A )
11
517The value of ( x ) between 0 and ( 2 pi ) which
satisfy the equation ( sin x sqrt{8 cos ^{2} x}=1 )
are in A.P. The common difference of the
A.P. is
A ( cdot frac{pi}{8} )
B. ( frac{pi}{4} )
c. ( frac{3 pi}{8} )
D. ( frac{5 pi}{8} )
11
518A chord of a circle of radius ( 12 mathrm{cm} )
stands an angle of ( 120^{circ} ) at the centre.
Find the area of the corresponding
segment of the circle.
11
519Prove the following identities:
( 1+cos ^{2} 2 x=2left(cos ^{4} x+sin ^{4} xright) )
11
52042. If the inequality sin²x + a cos x + a² > 1 + cos x holds for
any x e R, then the largest negative integral value of a is
a. 4
b. -3
c. -2
d. -1
11
521If ( sin ^{x} alpha+cos ^{x} alpha geq 1,0<alpha<frac{pi}{2}, ) then
A. ( x in[2,+infty) )
в. ( x in(-infty, 2] )
c. ( x in[-1,1] )
D. none of these
11
522Find the value of ( 4 cos 60^{circ}+2 sin 30^{circ} )
( mathbf{A} cdot mathbf{0} )
B . 2
( c .-3 )
D. 3
11
5233. The number of values of x in the interval [0,57] satisfying
the equation 3 sin’x – 7 sin x + 2 = 0 is
b. 5
d. 10 (IIT-JEE 1998)
a. 0
c. 6
11
524Prove that ( frac{1}{sin 10^{circ}}-frac{sqrt{3}}{cos 10^{circ}}=4 )11
525( sin theta=frac{3}{5}, operatorname{cosec} theta=? )11
526• The value of the expression (2 sin2 91°- 1) (2 sin? 92°- 1)
… (2 sind 180° – 1) is equal to
a. 0
d. 290 – 90
b. 1
c. 290
11
527Find general solution of the following equations:
( sin theta=frac{1}{2} ? )
11
528Solve: ( sqrt{sec x-1}=tan x )11
5291. The number of all the possible triplets (ay, ay, ay) such that
a, + a2 cos(2x) + az sin*(x) = 0 for all x is
a. O
b. 1
c. 3
d. infinite (IIT-JEE 1987)
11
530If ( sin A=x, cos B=y ) and ( A+B+ )
( C=0, ) then ( x^{2}+2 x y sin C+y^{2} ) is
equal to
( mathbf{A} cdot sin ^{2} C )
B. ( cos ^{2} C )
c. ( 1+sin ^{2} C )
D. ( 1+cos ^{2} C )
11
5314. If x, y e R satisfies (x + 5)2 + (y – 12)2 = (14)2, then the
minimum value of Vx² + y2 is_
2
11
532Prove that: ( cos ^{2} A+cos ^{2} B+cos ^{2} C= )
( 1-2 cos A cos B cos C )
11
5336. (a + 2) sin a + (2a – 1) cos a=(2a + 1) if tan a is
a. 3/4
b. 4/3
c. 2a/(a + 1)
d. 2al(a? – 1)
11
534If ( sec theta+tan theta=p )
Find ( csc =? )
11
535Illustration 4.9 Find the number of solutions of sinºx – sinx
– 1 = 0 in [-216, 210).
11
536( mathrm{IF} sin Theta=frac{8}{17} ) where ( 0^{circ}<Theta<90^{circ} )
then ( tan Theta+sec Theta ) is
( A cdot frac{1}{3} )
B. ( frac{2}{3} )
( c cdot frac{4}{3} )
D.
11
537The expression ( 2 cos 10^{circ}+sin 100^{circ}+ )
( sin 1000^{circ}+sin 10000^{circ} ) is simplified
then it simplifies to?
11
538( tan left(rho frac{pi}{4}right)=cot left(q frac{pi}{4}right) )
A ( . rho+q=0 )
В ( cdot rho+q=2 n+1 )
c. ( rho+q=2 n )
D ( cdot rho+q=2(2 n+1) )
11
539Convert ( frac{7 pi}{36} ) into degrees.11
540If ( tan theta=frac{1}{sqrt{5}} ) and ( theta ) lies in the first
quadrant, the value of ( cos theta ) is
A ( cdot frac{1}{sqrt{6}} )
B. ( frac{sqrt{5}}{sqrt{6}} )
c. ( frac{-1}{sqrt{6}} )
D. ( frac{-sqrt{5}}{sqrt{6}} )
11
5417. Show that 16cos( 25 cos ( 16 cos ( 15 cos 165 = 1
CC
(1983 – 2 Marks)
11
5425.
Let 2sin²x + 3sinx-2> 0 and x2-x-2<0 (x is measured in
radians). Then x lies in the interval
(1994)
(c) (-1,2)
11
543Q Type
elevation of the top ( P ) of the tower ( O P ) at
a point ( A ) on the ground is ( alpha, ) he then
walks a distance ( A B ) towards the foot ( O )
of the tower and finds the angle of
elevation as ( beta, ) he again walks a
distance ( B C ) in the same direction and
observes the angle of elevation now is ( gamma )
He notices that ( boldsymbol{a}+boldsymbol{beta}+boldsymbol{gamma}= )
( 180^{circ} ; alpha, beta, gamma ) are in A.P. and the distance
of ( C ) from the foot of the tower is half the
distance of ( B ) from the foot ( O ) of the
tower. If the height of tower is ( h ), the distance of ( A ) from the foot ( O ) is ( frac{5 h}{3 sqrt{3}} )
Reason

If the angles of elevation of the top of a
tower at three points on the ground are in A.P. then the distance of the points from the foot of the tower are also in A.P.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect

11
544( cos frac{pi}{7}+cos frac{2 pi}{7}+cos frac{3 pi}{7}+cos frac{4 pi}{7}+ )
( cos frac{5 pi}{7}+cos frac{6 pi}{7}+cos frac{7 pi}{7}= )
( A )
в.
c.
D.
11
54527. For x e(0,), the equation sinx +2sin 2x -sin 3x = 3 has
(JEE Adv. 2014)
(a) infinitely many solutions
(b) three solutions
(c) one solution
(d) no solution
11
54615.
Find the smallest positive number p for which the equation
cos( sin x) = sin(pcos x) has a solution x € (0.27)
(1995 – 5 Marks)
11
547Illustration 4.65
Solve cos 2x > sinxl, xe
Bla
11
548Illustration 4.1
Solve the equation sinx + cosx = 1.
11
549( frac{1}{sin ^{2} theta}-cot ^{2} theta ) is equal to
( mathbf{A} cdot mathbf{1} )
B. –
( c cdot 2 )
D. -2
11
550If ( sin theta=-0.6, ) the find the quadrant
from which the terminal arm making an
angle of ( theta^{circ} ) passes.
A . I quadrant
B. II quadrant
c. III quadrant
D. IV quadrant
11
5513. tan 100° + tan 125° + tan 100° tan 125° is equal to
b. 1/2
c. -1
d. 1
a. 0
11
552If ( cos A=frac{sqrt{3}}{2}, ) then ( tan 3 A= )
( mathbf{A} cdot mathbf{0} )
B. ( frac{1}{2} )
c. 1
( D cdot alpha )
11
553If ( tan ^{2} alpha=1-p^{2}, ) then ( sec alpha+tan ^{3} alpha )
( operatorname{cosec} alpha= )
11
55459. The general solution of cos x cos 6x = -1 is
a. x = (2n + 1) n, ne Z b. x = 2nt,n e Z
c. x = nt, n e Z d. none of these
d.
11
55575. The number of ordered pairs which satisfy the equation
x + 2x sin(xy) + 1 = 0 are (where y e [0, 21])
a. 1
b. 2
c. 3
d. O
11
5562. Iff (x) = 2(7 cos x + 24 sin x)(7 sinx – 24 cos x), for every
xe R, then maximum value of (f(x))/4 is
11
557Find the range of ( 3 cos theta-4 sin theta )11
558– VI
+ sin
+
5. In triangle ABC, prove that sin
Hence, deduce that
2
T + C
1
T + A
COS 4
+ B
COS 4
COS4
=;
11
559Illustration 2.48 Find the range of
f(x) = sin’x – 6sin x + 9+3.
11
560( A ) vertical tower stands on a declivity
which isinclined at ( 15^{0} ) to the horizon.
From the foot of the tower a man
ascends the declivity for80 feet and then finds that the tower subtends an
angle of ( 30^{0} . ) The height of the tower is
A ( cdot 20(sqrt{6}-sqrt{2}) f t )
B . ( 40(sqrt{6}-sqrt{2}) f t )
c. ( 40(sqrt{6}+sqrt{2}) f t )
D. ( 40 mathrm{ft} )
11
561If in a triangle ( A B C, sin A cos B=1 / 4 )
and ( 3 tan A=tan B, ) then the triangle
is
A. right angled
B. equilateral
c. isosceles
D. none of these
11
562Find the slope of the inclination of the
line of the following:
( boldsymbol{theta}=mathbf{3 0}^{circ} )
A. ( sqrt{3} )
B. ( frac{1}{sqrt{3}} )
c. ( frac{2}{sqrt{3}} )
D. ( frac{sqrt{3}}{2} )
11
563Arrange the following equations in decreasing order of their number of solutions in ( [mathbf{0}, mathbf{2} boldsymbol{pi}] )
I) ( 3 sin ^{2} theta+4 cos ^{2} theta=5 )
II) ( 4 sin ^{2} theta+3 cos ^{2} theta=frac{7}{2} )
|||| ( 3 mid sin ^{2} theta+4 cos ^{2} theta=4 )
A . ॥,|॥,
B. ।, ॥।, III
c. ॥ा,।, ॥
( mathbf{D} cdot||, ) И,
11
564Find ( x ) from the figure11
565( A=cos ^{2} theta+sin ^{4} theta, ) find range of ( A )11
566Prove that ( tan left(2 times 30^{circ}right)=frac{2 tan 30^{circ}}{1-tan ^{2} 30^{circ}} )11
567The angle subtended at the centre of
circle of radius 3 metres by an arc of length 1 metre is equal to
A ( cdot 20^{circ} )
B . ( 60^{circ} )
c. ( frac{1}{3} ) radian
D. 3 radian
11
5685. Number of solutions of equation 2 sin cos? x–2sin
sin? x = cos? x – sin? x for x € [0, 41] is
a. 6
b. 8
d. 12
c. 10
11
56910. Let A, B, C be three angles such that A = 7/4 and tan B
tan C = p. Find all possible values of p such that A, B, C
are the angles of a triangle.
11
570If ( tan 9^{circ}=frac{P}{q}, ) then the value of
( frac{sec ^{2} 81^{circ}}{1+cot ^{2} 81^{circ}} ? )
11
57112. If tan B =
nsin a cos a
, prove that tan(a – b) = (1 – n)
tan a
1- nsin’ a’
11
572Convert ( frac{3 pi}{7} ) in degrees.11
57330. If cos? x +
(1 + tan? 2y) (3 + sin 3z) = 4, then
cos- x)
a. x is an integral multiple of a
b. x cannot be an even multiple of a
c. z is an integral multiple of a
d. y is an integral multiple of /2
11
5742. Given A = sin²0+ costo, then for all real 0,
a. 1SAS2
b. 3/4 SAS1
c. 13/16 S AS1 d. 3/4 S A S 13/16
(UTLIFE 1080
11
575The value / values of ( x ) is/are
( mathbf{A} cdot pm 5 sqrt{5} )
B. ( pm sqrt{5} )
( ^{mathrm{c}}+frac{1}{sqrt{5}} )
D. none of these
11
576Illustration 2.3. Prove that
sec A-tan A
cOS A
COS A
sec A + tan A
11
577Illustration 2.57 Prove that
cos(90° + ) sec(-o) tan(180°-O)
-=
sec(360°-) sin(180°+2) cot(90°-0)
-1.
11
57817. In a triangle ABC, if A – B = 120° and sin –
, then the value of 8 cos C is
11
57937. The number of solutions of sec²0+ cosec?0+ 2 cosec?e
= 8,0 S OS Tt/2 is
a. 4
b. 3
c. O
d. 2
11
580Find all the values of ( theta ) which satisfy the equation: ( cos theta cdot cos 2 theta cdot cos 3 theta=1 / 4 )11
581If ( 270^{circ}<A<360^{circ}, 90^{circ}<B< )
( 180^{circ}, cos A=frac{5}{13}, tan B=-frac{15}{8}, ) then
( sin (boldsymbol{A}+boldsymbol{B})= )
A ( cdot frac{140}{221} )
в. ( frac{171}{221} )
c. ( frac{140}{171} )
D. ( frac{221}{171} )
11
5826. Let sin(@-a)
a
sin(0-3) b
a. cos (a-B)
c. sin (a + B)
cos(-a) C ac+bd
then
cos(0-B) d ad + bc
b. sin (a-B)
d. none of these
11
583The number of solutions of the equation ( x^{3}+2 x^{2}+5 x+2 cos x=0 ) in ( [0,2 pi] ) is:
A .
в.
( c cdot 2 )
D. 3
11
584When ( n ) is a even natural number then
the value of ( x ) is
This question has multiple correct options
A. 0
в. ( pi / 2 )
( mathrm{c} cdot 2 pi / 3 )
D. ( pi )
11
58510.
The
portion of a vertical pole subtends an
angle tan — at a point in the horizontal plane through its
foot and at a distance 40 m from the foot. A possible height
of the vertical pole is
[2003]
(a) 80m (b) 20 m (c) 40m (d) 60 m.
iges un and
11
58618. If sec x + sec? x =1 then the value of tan® x – tanº x –
2 tan- x + 1 will be equal to
a. 0
b. 1
c. 2
d. 3
11
58772. The number of solutions of the equation
12 sin x-5312cos -scOS = 1 in [0, 1] is
a. 2 to contatto b.
3
c. 4 O
d. 5
0
0
11
588Prove that: ( sin left(frac{pi}{2}-thetaright)=cos theta )11
5891. If sin = = and cos O=
=- V, then the general value of
O is (n e Z)
2
a. 2n +
b. 2nt +
BOB
c. 2nt +
d. 2nt +
11
59011. The number of solutions of the pair of equations
2 sin? 0 – cos 20= 0
2 cos0 – 3 sin 0 = 0
in the interval [0, 27] is
b. 1
c. 2
d. 4
(IIT-JEE 2007
a. 0
11
591prove that:
( frac{sin 5 x-2 sin 3 x+sin x}{cos 5 x-cos x}=tan x )
11
592Show that sin?5° + sinº 10° + sin? 15° + …
Illustration 2.60
+ sin 90º = 9-.
11
59367. The total number of solutions of sin {x} = cos {x} (where
{} denotes the fractional part) in [0, 2T] is equal to
a. 5
b.
61
c. 8
d. none of these
11
594Prove that ( 4left[sin ^{4} 30^{circ}+cos ^{4} 60^{circ}right]- )
( 3left[cos ^{2} 45^{circ}-sin ^{2} 90^{circ}right]=2 )
11
595The value of ( 144^{circ} ) in circular measure is
A ( cdot frac{3 pi^{c}}{4} )
B. ( frac{2 pi^{c}}{3} )
c. ( frac{4 pi^{c}}{5} )
D. ( frac{5 pi^{c}}{6} )
11
59613. Show that
sin x sin 3x
cos 3x cos 9x
cos 9x
cos 27x
[tan 27x – tan x].
11
597Illustration 2.30 Evaluate the sine, cosine, and tangent of
each of the following angles without using a calculator:
300°, -4050 71 117
11
5989. If x, y = [0, 21] and sin x + sin y = 2, then the value of
x + y is
a. T
b. 7/2
c. 310
d. none of these
11
599Evaluate the value of ( sin frac{pi}{6}+cos frac{pi}{6} )11
60010. Solve sinx+ sin ( 3 V1 –cos 2x)’ +sin?2x) = 0,11
601Illustration 2.5 Convert it/6 rad to degrees.11
602Let ( boldsymbol{f}(boldsymbol{x})=log left(log _{frac{1}{3}}left(log _{7}(sin boldsymbol{x}+boldsymbol{a})right)right) )
be defined for every real value of ( x, ) then
the possible value of ( a ) is This question has multiple correct options
( A cdot 7 )
B. 8
( c .9 )
D. 6
11
60327. Let f(x) = cos(a, + x) + -cos(az + x)+ 2 cos(az + x)
+…+ – cos (an + x) where aj, az, … ane R. If f(x)
= f(x) = 0, then (x2 – x; may be equal to
a.
T
b
.
211
c. 30
d.
1/2
11
604If ( 7 cos ^{2} theta+3 sin ^{2} theta=4 ) and ( theta ) is in first
quadrant .Show that ( boldsymbol{theta}=mathbf{6 0}^{circ} )
11
605If ( boldsymbol{A}=left{boldsymbol{x} epsilon boldsymbol{R} / frac{pi}{4} leq boldsymbol{x} leq frac{pi}{3}right} quad ) and
( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x}-boldsymbol{x} quad, ) then ( boldsymbol{f}(boldsymbol{A})= )
A ( cdotleft[frac{sqrt{3}}{2}-frac{pi}{3}, frac{1}{sqrt{2}}-frac{pi}{4}right] )
B. ( left[-frac{1}{sqrt{2}}-frac{pi}{4}, frac{sqrt{3}}{2}-frac{pi}{3}right] )
( mathbf{c} cdotleft[-frac{pi}{3}, frac{-pi}{4}right] )
D. ( left[begin{array}{c}pi \ hline 4end{array}, frac{pi}{3}right. )
11
606( frac{sin theta+sin 2 theta}{1+cos theta+cos 2 theta}=tan theta )11
607The value of
( cot 15^{circ} cot 20^{circ} cot 70^{circ} cot 75^{circ} ) is equal
to
A . -1
B. 0
( c .1 )
D. 2
11
608Assertion
The equation ( a sin x+cos 2 x=2 a-7 )
possesses a solution if ( boldsymbol{a} in[mathbf{2}, mathbf{6}] )
Reason
( -1 leq sin x leq 1 forall x in R )
A. Both Assertion and Reason are correct and Reason is
the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is
not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Both Assertion and Reason are incorrect
11
609( sec A=frac{12}{5} ) for some value of angle ( A )
A. True
B. False
11
610Find the value of ( csc 10^{circ}-sqrt{3} sec 10^{circ} )11
61116. If tan
*
+
+
prove that sin y – 3+sin²x
sin x
1+ 3 sin- x
11
612Convert ( frac{5 pi}{6} ) in to radians.11
613Match the following11
6144. The equation (cosp – 1)x2 + (cosp)x + sinp = 0 in the
variable x has real roots. Then p can take any value in the
interval
a. (0,21)
b. (- 1,0)
d. (0, 1)
(IIT-JEE 1990)
11
615( int tan ^{4} x sec ^{2} x d x )11
616Which of the following is correct (where ( boldsymbol{n} boldsymbol{epsilon} boldsymbol{N}) )11
617Prove that ( cos 40^{circ}+cos 50^{circ}+ )
( cos 70^{circ}+cos 80^{circ}=cos 20^{circ}+cos 10^{circ} )
11
618If ( alpha, beta ) are solutions of ( sin ^{2} x+ ) ( a sin x+b=0 ) and ( cos ^{2} x+c cos x+ )
( boldsymbol{d}=mathbf{0}, ) then ( sin (boldsymbol{alpha}+boldsymbol{beta}) ) equals
A ( cdot frac{2 a c}{a^{2}+c^{2}} )
B. ( frac{a^{2}+c^{2}}{2 a c} )
c. ( frac{2 b d}{b^{2}+d^{2}} )
D. ( frac{b^{2}+d^{2}}{2 b d} )
11
619If ( frac{sin A}{sin B}=p ) and ( frac{cos A}{cos B}=q, ) find ( tan A )
and ( tan B )
11
620Illustration 3.7
Illustration 3.7 If sin(A – B) – Tro.cos(A+B)- 20 find
If sin(A-B) =
find
the value of tan 2 A where A and B lie between 0 and /4.
11
62113. If 0 SX S27, then 2cosecºx – y – y+1 s 12
a. is satisfied by exactly one value of y
b. is satisfied by exactly two value of x
c. is satisfied by x for which cosx = 0
Cod. is satisfied by x for which sinx = 0
11
622Find the value of ( cos 75^{circ} )11
6232. The general solution of the trigonometric equation
sinx + cos x = 1 is given by
a. x = 2nt, n= 0, #1, #2, …
b. x = 2n1 + Tt/2; n= 0, +1, +2, …
c. x = nt+ (-1)” ?-?n=0, #1, #2, …
4 4 4
d. none of these
(IIT-JEE 1981)
11
624then prove that
Illustration 3.10 If a, b, ye 0,-
sin(a+B+y)
sin a +sin ß + sin y
-<1.
11
625w
1. The expression sin“ (3x –a)+ sin“Gr + c
-2 sin* (3x+a)=sin(57 –is equal to
a. 0
c. 3
b. 1
d. none of these
11
626Illustration 3.36
In quadrilateral ABCD, if
GOS
COS
then find the value of sin
sin — sin — sin
la
11
627Find the maximum value of ( sqrt{3} sin x+ )
( cos x )
11
62810. Number of roots of the equation
SUN
NIS
tan
x-
12
4
) 2(0.25) cos 2x + 1 = 0 is
11
629TTC
16.
Find all values of in the interval
satisfying the
2
equation (1 – tan 0) (1 + tan o) sec2
+ 2 tan e=0.
(1996 – 2 Marks)
11
630llustration 4.6
Solve 16sinx + 16c0s? x = 10,0 5x< 27.
11
631+ 2 cot
771
44. The value of cot-
16
a. 4
c. -2
157
+ cot –
16
51007
b. 2
d. -4
11
632A pole of height 20 ft has a shadow of length ( 11.55 mathrm{ft} ) at a particular instant of
time. Find the angle of elevation (in degree) of the sun at this point of time.
( mathbf{A} cdot 90 )
B. 60
( c cdot 30^{circ} )
D. 45
11
633COS
Illustration 3.38 In AABC, sin C + cos C + sin(2B + C) –
cos (2B + C) = 2V2. Prove that AABC is right-angled
isosceles.
11
634The value of ( cos left(-1044^{circ}right) ) is ( frac{(sqrt{5}+1)}{4} )
A . True
B. False
11
63596. 180sxs then range of fw) = sec (*_*)+ see(++)
sec
11
636Simplify, using trigonometric tables ( sin 50^{circ} 26^{prime}+cos 18^{circ}+tan 70^{circ} 12^{prime} )11
63719. If cotx = cot(x – y) cot(x – z), then cot2x is equal to
(where x #T /4)
a.
(tan y + tan z)
b. =(cot y+cot z)
.
.
.
– (sin
(sin y+sin z)
d. none of these
C
D
11
638( ln a Delta A B C, cos B cdot cos C+ )
( sin B cdot sin C cdot sin ^{2} A=1 . ) Then the
triangle is
A. right – angled isosceles
B. isosceles whose equal angles are greater then ( pi / 4 )
c. equilateral
D. none of these
11
6397. Show that 16008 ()cos ( 15 cos 15 cos 1657) – 1.11
640The maximum valuue of the expression ( frac{1}{sin ^{2} theta+3 sin theta cos theta+5 cos ^{2} theta} )11
641If ( sin ^{-1} x=y, ) then
A. ( 0 leq y leq pi )
B. ( -frac{pi}{2} leq y leq frac{pi}{2} )
c. ( 0<y<pi )
D. ( -frac{pi}{2}<y<frac{pi}{2} )
11
642Express the following angle in terms of first-quadrant reference angle:
( tan 336^{circ} )
A . ( tan 45 )
B . ( -tan 36 )
c. ( -tan 24 )
D. ( tan 24 )
11
643( a=operatorname{cosec} 2^{0}, b=operatorname{cosec} 2, ) then which of
the following holds true?
( mathbf{A} cdot a=b )
В ( . a>b )
c. ( a<b )
D. ( a=2 b )
11
644If ( tan theta=frac{sin alpha-cos alpha}{sin alpha+cos alpha}, ) then
( A cdot sin alpha-cos alpha=pm sqrt{2} sin theta )
B. ( sin alpha+cos alpha=pm sqrt{2} cos theta )
( mathbf{c} cdot cos 2 theta=sin 2 alpha )
( D cdot sin 2 alpha+cos 2 alpha=0 )
11
645Find the following:
( sin left[frac{pi}{2}-sin ^{-1}left(frac{-sqrt{3}}{2}right)right] )
11
646and
Illustration 3.82 In triangle ABC, if cot A. cot C = =
cot B.cot C = =, then the value of tan C is
11
647( ln Delta A B C, ) a ( sin (B-C)+b sin (C- )
( A)+c sin (A-B)= )
( A )
B. ( a+b+c )
c. ( a^{2}+b^{2}+c^{2} )
D ( cdot 2left(a^{2}+b^{2}+c^{2}right) )
11
648The number of solutions of ( sin 3 x= )
( cos 2 x, ) in the interval ( left(frac{pi}{2}, piright) ) is
( A cdot 3 )
B. 4
( c cdot 2 )
D.
11
649If ( sin A+cos A=sqrt{2} ) then ( sin A )
( cos A ) is equal to:
11
650Find the value of ( operatorname{cosec}^{2} 45^{circ}-cot ^{2} 45^{circ} )11
651If ( operatorname{Sin}^{2} A+operatorname{Cos}^{2} B+operatorname{Sin}^{2} C=1, ) then
the triangle ( A B C ) is
A. isosceles
B. equilateral
c. right angled
D. scalene
11
652Illustration 3.76 Prove that in triangle ABC, cos?A + cos²B
– cos C = 1 – 2 sin A sin B cos C.
11
653If ( cos ^{2} x=cos ^{4} x ) then find the
minimum and maximum value of
( sin ^{2} x+cos ^{4} x, ) for all real values of ( theta )
11
654Degree measure of ( left(frac{7 pi}{6}right)^{c} ) is
A ( .210^{circ} )
B . ( 240^{circ} )
( c cdot 270 )
D. None of these
11
6553. IfK=sin(īt/18) sin(57/18) sin(77/18), then the numerical
value of K is
(IIT-JEE 1993)
11
65613. The value of f(a) Vcosec’a – 2 cot a
+Vcoseca – 2 cot a can be
a. 2 cot a
b. – 2 cot a
c. 2
b. -2
11
657Illustration 3.30
Prove that cos 18º – sin 18° = 12 sin 27°
11
658( frac{cos ^{2} 30^{circ}+cos 30^{circ} sin 30^{circ}+sin ^{2} 30^{circ}}{cos ^{3} 30^{circ}-sin ^{3} 30^{circ}} )11
659The value of ( 1+cot ^{2} A ) is
( mathbf{A} cdot cos ^{2} A )
B. ( sec ^{2} A )
( mathbf{c} cdot tan ^{2} A )
D. ( operatorname{cosec}^{2} A )
11
660( sin ^{3} x-sin ^{3}left(240^{0}-xright)+sin ^{3}left(240^{0}+right. )
( boldsymbol{x}) in )
( [-boldsymbol{k}, boldsymbol{k}]=rangle boldsymbol{k}= )
( A )
B. ( 1 / 4 )
( c cdot 3 / 4 )
( D cdot 5 / 4 )
11
661Illustration 2.61 Ifsin(120°-a)=sin(120°– B), 0<a, ß<TT,
then find the relation between a and B.
11
662If ( tan A+sec A=3, ) Find the value of ( sin )
( mathbf{A} )
11
663( frac{sin (boldsymbol{A}-boldsymbol{B})}{cos boldsymbol{A} cos boldsymbol{B}}+frac{sin (boldsymbol{B}-boldsymbol{C})}{cos boldsymbol{B} cos boldsymbol{C}}+ )
( frac{sin (boldsymbol{C}-boldsymbol{A})}{cos boldsymbol{C} cos boldsymbol{A}}=boldsymbol{0} )
11
66417. If 5(tan?x-cos2x)=2cos 2x +9, then the value of cos 4x is:
[JEEM 2017
11 L
lations of the equati
11
6658. If cos 4x = a + a,cos²x + a cosx is true for all
values of x e R, then the value of 5a, + aj + az
11
66622.
Given both O and o are acute angles and sin
=
cos o = =, then the value of 0 + 0 belongs to
(2004)
11
66760. The number of solutions the equation cos(O). cos(110) = 1
has
b. 2 O
c. 1
d. infinite
a. 0
11
668Find general solution for ( x ) ( sin 8 x-cos 6 x=sqrt{3}(sin 6 x+cos 8 x) )11
669Illustration 4.55
Solve cos 40+ sin 50= 2.
11
67012. The value of tan?
tan? 24 tan2 341 is
a. – 3
c. -5
b. 7
d. none of these
11
671What is the greatest value of ( theta ) lying between ( 0^{0} ) and ( 720^{0} ) whose tangent is ( -frac{1}{sqrt{3}} ? )
A ( cdot 690^{circ} )
( ^{0} )
в. ( 510^{circ} )
c. ( 330^{circ} )
D. ( 150^{circ} )
11
672If ( cos A=frac{4}{5}, cos B=frac{12}{13}, frac{3 pi}{2}<A, B< )
( 2 pi, ) find the values of the following.
( (i) cos (A+B) )
(ii) ( sin (A-B) )
11
673The value of ( sec ^{2} A tan ^{2} B- )
( tan ^{2} A sec ^{2} B ) is
( mathbf{A} cdot tan ^{2} B-tan ^{2} A )
B ( cdot sec ^{2} B+sec ^{2} A )
c. ( tan ^{2} B-sec ^{2} A )
( mathbf{D} cdot sec ^{2} B-tan ^{2} A )
11
674A wheel makes 12 revolutions per hour
The radians it turns through in 20
minutes is:
( A cdot 8 pi^{c} )
В . ( 16 pi^{c} )
( c cdot 24 pi^{c} )
D. ( 32 pi^{c} )
11
675If ( tan A=frac{1}{2} ) and ( tan B=frac{1}{3}, ) then the
value of ( boldsymbol{A}+boldsymbol{B} ) is
A ( cdot frac{pi}{6} )
в. ( pi )
c. zero
D.
11
676Solve: ( cos ^{2} x-2 cos x=4 sin x- )
( sin 2 x, 0 leq x leq pi )
11
67714. Determine the smallest positive value of x (in degrees) for
which
tan(x +100°)=tan(x +50°) tan(x) tan (x – 50°).
(1993 – 5 Marks)
11
678If ( tan (A-B)=1, sec (A+B)=frac{2}{sqrt{3}} )
then the smallest +ve value of B is?
A ( cdot frac{25 pi}{24} )
в. ( frac{19 pi}{24} )
c. ( frac{13 pi}{24} )
D. ( frac{11 pi}{24} )
11
679What is the reference angle, ( rho ? )
( mathbf{A} cdot rho=30^{circ} )
( mathbf{B} cdot rho=45^{circ} )
( mathbf{C} cdot rho=56^{circ} )
D ( cdot rho=270^{circ} )
11
6802. The number of values of 0 in the interval
(
NT
such that ex mat for n= 0, +1, + 2 and
such that 0 + – for n = 0,
I 22
tan O=cot 50 as well as sin 20 = cos 40 is
1,
2 and
TOO
11
681Illustration 4.42 Solve 4 cot20= cote- tane.11
682Illustration 4.44
Solve 73 cos 0+ sin 0 = V2.
11
683Illustration 4.51 If 3 sinx + 4 cos ax = 7 has at least one
solution, then find the possible values of a.
11
684If ( mathbf{0} leq boldsymbol{x} leq mathbf{3} boldsymbol{pi}, mathbf{0} leq boldsymbol{y} leq )
( 3 pi ) and ( cos x . sin y=1 ) then the
possible number of values of the
ordered pair ( (x, y) ) is
( A cdot 6 )
B. 12
c. 8
D. 15
11
6856.
If

X
tan ay
, then show that
tan(6+) tan(8+B) tan(6+y)’
2 * sin?(a – B)=0.
x – y
11
686Which of the following do/does not
reduce to unity?
This question has multiple correct options
A ( frac{sin left(180^{0}+Aright)}{tan left(180^{0}+Aright)} frac{cot left(90^{0}+Aright)}{tan left(90^{0}+Aright)} frac{cos left(360^{0}-Aright) operatorname{cosec} A}{sin (-A)} )
B. ( frac{sin (-A)}{sin left(180^{0}+Aright)}-frac{tan left(90^{0}+Aright)}{cot A}+frac{cos A}{sin left(90^{0}+Aright)} )
C. ( frac{sin 24^{0} cos 6^{0}-sin 6^{0} cos 24^{0}}{sin 51^{0} cos 69^{0}-cos 51^{0} sin 69^{0}} )
D. ( frac{cos left(90^{0}+Aright) sec (-A) tan left(180^{0}-Aright)}{sec left(360^{0}+Aright) sin left(180^{0}+Aright) cot left(90^{0}-Aright)} )
11
687Which of the following is correct:
( A cdot sin 1^{circ}>sin 1 )
B. ( sin 1^{circ}<sin 1 )
( mathbf{c} cdot cos 1^{circ}<sin 1 )
D. None of these
11
688The value of ( sin 105^{0} ) is
A ( cdot frac{sqrt{3}-1}{2 sqrt{2}-1} )
B. ( frac{sqrt{3}-1}{sqrt{2}-1} )
c. ( frac{sqrt{3}+1}{2 sqrt{2}} )
D. ( frac{sqrt{3}+1}{sqrt{2}} )
11
689Illustration 2.56 Prove that sin(-420°) (cos 390°) +
cos (-660°) (sin 330°) = -1.
11
69012. If cos(x + 7/3) + cos x = a has real solutions, then
a. number of integral values of a are 3
b. sum of number of integral values of a is 0
c. when a = 1, number of solutions for xe [0, 276] are
3
d. when a = 1, number of solutions for xe [0, 2T) are
11
691Prove that
( frac{cos e c A+1}{cos e c A-1}=frac{1+sin A}{1-sin A} )
11
692Prove that:
( sin ^{6} theta+cos ^{6} theta+3 sin ^{2} theta cos ^{2} theta=1 )
11
693If ( boldsymbol{A}+boldsymbol{B}+boldsymbol{C}=boldsymbol{pi}, ) then prove the
following ( sin 2 A+sin 2 B+sin 2 C= )
( 4 sin A cdot sin B cdot sin C )
11
694sin- – cos3
Solve –
2 + sin x
2
COS X
Illustration 4.20
11
695Find the value of ( boldsymbol{theta}, ) if
( cos theta=0.9664 )
11
696Illustration 4.43 Find the most general solution of
21+ |cos x + cos x + |cosxp + … =4
11
69721. Let S = {0€[-21, 21t]: 2 cos 0+3 sino=0}.
Then the sum of the elements of S is:
[JEEM 2019-9 April (M
11
69812.
If exp {(sinx + sin^x + sinºx + …………… (0) In 27
satisfies the equation x2- 9x + 8 = 0, find the value of
| cosx 0<x< ".
(1991 – 4 Marks)
cos x + sinx
com <**
29. septes
11
699Given ( boldsymbol{A}=sin ^{2} boldsymbol{theta}+cos ^{4} boldsymbol{theta}, ) then for all
real ( theta, ) which of the following is true?
A. ( 1 leq A leq 2 )
в. ( frac{3}{4} leq A leq 1 )
c. ( frac{13}{16} leq A leq 1 )
D. ( frac{3}{4} leq A leq frac{13}{16} )
11
700Find the value of ( x ) if ( sin ^{-1}left(frac{2}{3}right)+ ) ( sin ^{-1}left(frac{2}{3}right)=sin ^{-1} x )11
701if ( sin theta=frac{1}{2} ) and ( cos theta=-frac{sqrt{3}}{2}, ) then
the general value of ( theta i s(n epsilon z) )
A ( cdot 2 n pi+frac{5 pi}{6} )
В ( cdot 2 n pi+frac{pi}{6} )
c. ( _{2 n pi+frac{7 pi}{6}} )
D. ( 2 n pi+frac{pi}{4} )
11
70272. The value of tan 9° – tan 27° – tan 63° + tan 81° is
a. 2
b. 3
c. 4
d. none of these
11
703In a triangle ( A B C, ) the least value of ( tan ^{2} frac{A}{2}+tan ^{2} frac{B}{2}+tan ^{2} frac{C}{2} ) is
( mathbf{A} cdot mathbf{1} )
B.
( c cdot sqrt{3} )
D. ( frac{2}{3} )
11
7047
63. If cos 3x + sin 2x –
– 2, then x is equal to (ke Z)
6
a. (6x +1)
ok +1
b. (k-1)
(2k +1)
d. none of these
11
705The value of ( sec ^{3} theta, ) where ( theta ) is the acute
angle between the plane faces of a regular tetrahedron, is
11
706( ln Delta A B C, ) if ( 2 sin ^{2} B+2 sin ^{2} C+ )
( cos 2 A=1 ) then the triangle is
A. right angled
B. isosceles
c. equilateral
D. right angled and isosceles
11
707( sin frac{7 pi}{4}=frac{1}{sqrt{2}} )
A. True
B. False
11
708, where x e 1st quadrant, then tan-
62. If sin x + cos x =
is equal to
2
7-2
d. none of these
11
709( ln a Delta A B C, ) prove that
( (a) cos (A+B)+cos C=0 )
(b) ( tan frac{(A+B)}{2}=cot frac{C}{2} )
11
710( sin 20^{0} times cos 40^{0} times tan 45^{0} times cos 90^{0}= )11
7116. Number of solutions of the equation 4(cos22 x + cos2x+1)
+ tan x(tan x-2/3) = 0) in [0, 21] is
a. 0
b. 1
c. 2
d. 3
11
71228. If 0 < x < 21, then the number of solutions of 3(sin x +
cosx) – 2(sinx + cosx) = 8 is
a. 0
b. 1
c. 2
due to di
11
713Illustration 2.49 If (x, y) satisfies the equation 1 + 4x – x2
= 19 sec? y + 4 cos ec?y, then find the value of x and tan”y.
11
714coto
Sa
16. Let f(0) =
16. Let f(0) = 1 + cote and a + ß
, then the value of
4 , then ti
f(a) f(B) is
b.

2
c. 2
d. none of these
11
715The possible value(s) of ‘ ( boldsymbol{theta} ) ‘ satisfying
the equation ( sin ^{2} theta tan theta+ )
( cos ^{2} theta cot theta-sin 2 theta=1+tan theta+ )
( cot theta, ) where ( theta in[0, pi] ) is/are
A.
в. ( pi )
c. ( frac{7 pi}{12} )
D. None of these
11
71626. The least value of 2 sin’e + 3 coso is
a. 1
b. 2
c. 3
d. 5
11
717Illustration 3.27 If sin?(0 – a) cos a = cos(-a)sin a =
m sin a cos a, then prove that m| 3 –
11
718Solve
( tan theta=frac{sin alpha-cos alpha}{sin alpha+cos alpha} )
11
719In triangle ( A B C, ) right-angled at ( B ), if ( tan A=frac{1}{sqrt{3}} ) find the value of:
(i) ( sin A cos C+cos A sin C )
11
72085. The equation tan*x – 2 sec²x + a=0 will have at least one
solution if
a. 1<a s4
b. a 2
c. a <3 O
d
d. none of these
none
11
721If ( sqrt{3} tan theta=3 sin theta, ) find the value of
( sin ^{2} theta-cos ^{2} theta )
11
7226. Convert 18 degree into radians.
rad
rad
11
723Find the radian measure corresponding
to the degree ( -47^{circ} 30^{prime} )
A ( cdot frac{-19 pi}{72} r a d )
В ( cdot frac{19 pi}{72} r a d )
c. ( frac{13 pi}{72} ) rad
D. None of these
11
724The value of ( cos ^{2} 45^{circ}-sin ^{2} 15^{circ} ) is
( A cdot frac{3}{4} )
B. ( frac{sqrt{5}}{4} )
( c cdot frac{sqrt{3}}{4} )
D. ( frac{sqrt{5}+1}{4} )
11
725Express in degree form ( left(frac{5 pi}{12}right)^{c} )
A ( .60^{circ} )
B . 45
( c cdot 180 )
D. ( 90^{circ} )
11
726The number of solutions of the equation ( mathbf{1}+sin boldsymbol{x} cdot sin ^{2} frac{boldsymbol{x}}{mathbf{2}}=mathbf{0}[-boldsymbol{pi}, boldsymbol{pi}] ) is
A . zero
B.
( c cdot 2 )
( D )
11
727Convert ( 40^{circ} 20^{prime} ) into radian measure.
A ( cdot frac{121}{540} pi ) radians
в. ( frac{121}{570} pi ) radians
c. ( frac{120}{513} pi ) radians
D. None
11
728is
1. If 0 Sost and 81sin’e + 81c0s? 0 = 30, then
a. 30°
b. 60°
c. 120°
d. 150°
11
729Illustration 4.27 Solve 5 cos 20 + 2 cos2 – +1 = 0,
– 1<<.
11
730If ( sin 25^{circ} cdot sin 35^{circ} cdot sin 85^{circ}=frac{cos x^{circ}}{a} ) wher
argument of ( cos ) is acute and positive then find the value of ( (x+a) )
11
731What will be the values of ( theta ) between ( 0^{circ} )
and ( 360^{circ} ) if ( sin theta=-frac{sqrt{3}}{2} )
A ( cdot 30^{circ}, 330^{circ} )
В. ( 60^{circ}, 120^{circ} )
c. ( 135^{circ}, 315^{circ} )
D . ( 240^{circ}, 300^{circ} )
11
732If ( frac{x}{cos theta}=frac{y}{cos left(theta-frac{2 pi}{3}right)}= )
( frac{z}{cos left(theta+frac{2 pi}{3}right)}, ) then ( x+y+z ) is equal to
( A )
B.
( c cdot-1 )
D. none of these
11
733Degree measure of ( left(frac{1}{4}right)^{c} ) is
A ( cdot 15^{circ} 19^{prime} 5^{prime prime} )
B . ( 14^{circ} 19^{prime} 5^{prime prime} )
( mathbf{c} cdot 15^{circ} 18^{circ} 6 ” )
D. ( 14^{circ} 18^{circ} 6 ” )
11
734Find the value of ( left(4 cdot sin 30^{0} cos 30^{0}right)^{2} )11
735The value of ( 60^{circ} ) in centesimal system is
A ( cdot frac{100^{9}}{3} )
в. ( frac{200^{9}}{3} )
c. ( frac{140^{9}}{3} )
D. ( frac{160^{9}}{3} )
11
736Express 1.2 rad in nereast integer
degree measure.
11
7371. If tan 0+ sin 0 = m and tan 0 – sin 0= n, then
a. m² – n² = 4mn b. m² + n² = 4mn
c. m – n2 = m² + 12 d. m – n2 = 47mn
11
738( sin theta=cos theta ) for all values of ( theta )
Enter 1 for true and 0 for false
11
739In a triangle ( A B C )
( boldsymbol{b} cos (boldsymbol{C}+boldsymbol{theta})+boldsymbol{c} cos (boldsymbol{B}-boldsymbol{theta})= )
A ( . a cos theta )
B. ( a sin theta )
c. ( a tan theta )
D. a cot ( theta )
11
740Given: ( cos A=0.6 ; ) find all other
trigonometrical ratios for angle ( boldsymbol{A} )
11
74111.
For O<O<
the solution (s) of
m
Cosec
cosec
m
=1
is (are)
(2009)
11
742Solve ( left(sec ^{2} theta-1right)left(1-operatorname{cosec}^{2} thetaright)=-1 )11
743Radian measure of ( 175^{circ} 45^{prime} ) is
A ( cdot frac{700}{720} pi )
в. ( frac{703}{720} pi )
c. ( frac{705}{720} pi )
D. ( frac{710}{720} pi )
11
744Solve the equation ( : cot x-2 sin 2 x=1 )
A ( cdot x=frac{pi}{8}+frac{K pi}{2} )
в. ( x=frac{pi}{4}+frac{K pi}{4} )
c. ( _{x}=frac{pi}{6}+frac{K pi}{3} )
D. ( x=frac{2 pi}{3}+frac{K pi}{3} )
11
745( frac{tan theta}{left(1+tan ^{2} thetaright)^{2}}+frac{cot theta}{left(1+cot ^{2} thetaright)^{2}}=sin theta )
( cos theta quad, ) then ( theta ) is
( mathbf{A} cdot 2 pi )
в. ( frac{pi}{56} )
( c cdot frac{pi}{2} )
D.
11
746Ilustration 2.54
Prove that sin
<
< tan
for OE (0, 70/2).
11
747( pi^{c} )
( frac{pi}{5} ) in sexagesimal measure is
A ( cdot 18^{circ} )
B. ( 36^{circ} )
( c cdot 54^{circ} )
D. 72
11
748Illustration 4.29
cos?x.
Solve the equation
sinx – cos x =
11
74966. If cos 2x = 1, the value of cos x
is (0° S XS 45°)
(1) 2
(2) O
(3) 1
(4) 2
000
e) ond
11
750It is known that ( sin beta=frac{4}{5} & 0<beta<pi )
then the value of
( frac{sqrt{mathbf{3}} sin (boldsymbol{alpha}+boldsymbol{beta})-frac{mathbf{2}}{mathbf{c o s}} frac{boldsymbol{pi}}{mathbf{6}} cos (boldsymbol{alpha}+boldsymbol{beta})}{sin boldsymbol{alpha}} )
A. independent of ( alpha ) for all ( beta ) in ( (0, pi) )
В. ( frac{5}{sqrt{3}} ) for ( tan beta0 )
( D . ) none
11
75110. Let x = sin 1°, then the value of the expression
11
cos 0°.cos 1° cos1º.cos 2° cos 2º.cos 3°
1
xord)
-+-
cos 44º. cos 450 is equal to
a. x
c. √21x
b. 1/x
d. x/ 12
11
752If ( frac{sin x}{1+sec x}+frac{sin x}{sec x-1}=2 ) where
( 0^{circ}<x<90^{circ} ) then cosec ( x ) has the value
equal to
( mathbf{A} cdot mathbf{1} )
B. 2
c. ( sqrt{2} )
D. ( sqrt{3} )
11
753Show that ( cot 7 frac{1^{circ}}{2}=sqrt{2}+sqrt{3}+sqrt{4}+ )
( sqrt{6} )
11
754Illustration 3.52 Find the least positive value of x satisfying
sin22x + 4 sin^x- 4 sinx cos²x 1
4-sin2x – 4 sin²x
11
755Illustration 2.47 Find the range of f(x) = sin?x – 3 sinx +211
756Prove that: ( frac{1}{operatorname{cosec} A-cot A}-frac{1}{sin A}= )
( frac{1}{sin A}-frac{1}{operatorname{cosec} A+cot A} )
11
757If ( sin (A+B)=1 ) and ( cos (A-B)= )
( frac{sqrt{3}}{2} . ) then find the minimum positive
value of ( boldsymbol{A} ) and ( boldsymbol{B} ) is
A ( cdot 60^{circ}, 30^{circ} )
( ^{circ} 0^{circ} )
B . ( 75^{circ}, 15^{circ} )
( mathbf{c} cdot 45^{circ}, 60^{circ} )
D. ( 45^{circ}, 45^{circ} )
11
758TT
Illustration 3.106 Let a, b, y> 0. and a +B+ y = 5. Then
prove that tan a tan B + ytan ſtan y + tan a tan y s 13
11
759The domain of the function ( y= )
( sqrt{sin x+cos x}+sqrt{7 x-x^{2}-6} ) is
( left[p, frac{q pi}{4}right] cupleft[frac{r pi}{4}, sright] ) then value of ( p+ )
( boldsymbol{q}+boldsymbol{r}+boldsymbol{s} ) is
11
760Simplify ( sqrt{2+sqrt{2+2 cos left(pi+60^{0}right)}} )11
761(1994)
10. Let 0<x< then (sec2x – tan2x) equals
an
X-
11
762Illustration 3.77 In triangle ABC, prove that
sin(B + C – A) + sin(C + A – B) + sin(A + B-C) = 4 sin A
sin B sin C.
11
763Prove that,
( cos 20^{circ} cos 40^{circ} cos 60^{circ} cos 80^{circ}=frac{1}{16} )
11
764The value of ( 45^{circ} ) in centesimal system
is
A ( cdot 25^{9} )
B. ( 50^{text {9 }} )
( mathrm{c} cdot 75^{9} )
D. ( 100^{9} )
11
765If x2 + y2 = 4, then find the maximum value
Illustration 3.94
of x² + y²
xty
11
766COS X
COS X
5. If cos(x – y), cosx and cos(x + y) are in H.P., then cos x
(IIT-JEE 1997)
Sec
11
767Assertion
The minimum value of the expression
( sin alpha+sin beta+sin gamma ) where ( alpha, beta, gamma ) are
real number such that ( boldsymbol{alpha}+boldsymbol{beta}+boldsymbol{gamma}=boldsymbol{pi} )
is negative because-
Reason
( alpha, beta, gamma ) are angles of a triangle.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
11
768If ( tan theta+sec theta=2 x ) then prove that
( sec theta=x+frac{1}{4 x} )
11
769In which quadrant ( 196^{circ} ) lies?
B. quadrant III
c. quadrantı
D. quadrant IV
11
7704. The real roots of the equation cos’x + sin x = 1 in
the interval (-1, T) are

–, and
11
771One root of the equation ( cos x-x+ )
( frac{1}{2}=0 ) lies in the interval?
A ( cdotleft(0, frac{pi}{2}right) )
В. ( left(-frac{pi}{2}, 0right) )
c. ( left(-frac{pi}{2}, piright) )
D. ( left(pi, frac{3 pi}{2}right) )
11
77222. One of the general solutions of 4 sin o sin 20 sin 40 =
sin 30 is
a. (3n +1) 6/12, Vne Z
b. (4n+1) 7/9, ne z
c. (3n+1) 7/9, ne z
d. (3n+1) 7/3, ne z
11
773Sum of all the values of ( x ) in the interval
( [0,100 pi] ) satisfying the equation
( sin x=0 ) is
( mathbf{A} cdot 4950 pi )
B. ( 5050 pi )
c. ( 5151 pi )
D. none of these
11
774Illustration 2.45
Find the range of f(x) =
5 sin x
6
11
77510. If sec4 e + sec? 0 = 10 + tanº e + tan’ 6, then sin? 0 =
Alw
wilt WIN
d.
alu
11
776Number of solations of the equation (1993 – I Mark)
wax* x2cos x lying in the interval [0,21] is:
R 0 (0) 1 (c) 2
(d) 3
11
77719. sing
become on
79. sinx + cos x = y2 – y + a has no value of x for any value
of y if a belongs to
a. (0, 13) b. (-13,0)
c. (- 0o – √3) d. (√3,0)
11
778( cos ^{2} A+cos ^{2}left(120^{circ}+Aright)+ )
( cos ^{2}left(120^{circ}-Aright)= )
A ( cdot frac{3}{2} )
в. ( frac{1}{2} )
( c cdot frac{3}{18} )
D.
11
779Illustration 3.102 In AABC, prove that
cosee 4 + cosec + cosce ©26.
cosec
+ cosec
— + cosec
NI
11
780If ( sin alpha=frac{1}{2}, ) then find value of ( 3 sin alpha- )
( 4 sin ^{3} alpha )
11
781Illustration 2.38
a. sin 3
c. sin 1
Which of the following is the least?
b. sin 2
d. sin 7
11
78215. Let a and B be non-zero real numbers such that
2(cosß- cosa) + cosa cosß= 1. Then which of the following
is/are true?
(JEE Adv. 2017)
11
783Prove the following identities:
( sin 3 x+sin 2 x-sin x= )
( 4 sin x cos frac{x}{2} cos frac{3 x}{2} )
11
78419.
is equal to
2r sin a
1 + 2r cos a
a. tan²0
c. cot 20
b. cote
d. tan 20
11
785Show that
( tan 36^{circ} tan 17^{circ} tan 54^{circ} tan 73^{circ}=1 )
11
78687. If a, b e [0, 21) and the equation x2 +4 + 3 sin(ax + b) –
2x = 0 has at least one solution, then the value of (a + b)
can be
not.
b. Sy
d. none of these
11
78730. The number of solutions of the equation cos 6x + tan²x +
cos 6x . tan²x = 1 in the interval [0, 21) is
b. 5
od = c.
6 20
d. 7
A
a. 4
11
788( cos frac{pi}{5} cos frac{2 pi}{5}=frac{1}{4} ) and ( cos frac{pi}{5}+ )
( cos frac{3 pi}{5}=frac{1}{2} ) without using standard
value
11
789If ( tan theta+tan 2 theta+tan theta tan 2 theta=1 ) then
general value of ( boldsymbol{theta} ) is
( mathbf{A} cdot n pi ; n in I )
в. ( n pi pm frac{pi}{3} ; n in I )
c. ( frac{n pi}{3}+frac{pi}{12}, n in I )
D. None of these
11
790If ( sec ^{2} theta=frac{4}{3}, ) then the general solution
of ( boldsymbol{theta} ) is
( mathbf{A} cdot 2 n pi pm pi / 6 )
в. ( 2 n pi pm pi / 3 )
c. ( n pi pm pi / 6 )
D. ( n pi pm pi / 3 )
11
7911. The positive integer value of n> 3 satisfying the equation
+-
is
(IIT-JEE 2011)
( 377
sin
sin
27

n
)
sin/
In
11
79214. Solve the equation sin’x cos 3x + cos’x sin 3x + 3 = 0.11
79371. If sin 0 + sinº0 = 1, then the val-
ue of cos 12 + 3 cos 10 m+ 3 cos8
0 + cos6 0 – 1 is
(1) 2
(2)

(3) 12
(4) O
11
794If ( sin alpha+sin beta=frac{1}{2} ) and ( cos alpha+ )
( cos beta=frac{sqrt{3}}{2}, ) then value of ( 3 alpha+beta ) is
A ( .90^{circ} )
B . ( 0^{circ} )
( c cdot 120^{circ} )
D. ( 60^{circ} )
11
79515. If cos 28° + sin 28° = k”, then cos 17° is equal to
the core 2 is equal to
k3
b. —
* Be
121 enero
V2
si
d. none of these
11
796The value of ( sqrt{left(frac{1+cos theta}{1-cos theta}right)}-csc theta )
( cot theta= )
11
797Match the statements of column ( I ) with
values of column ( boldsymbol{I} boldsymbol{I} )
11
79888. The sum of all roots of sin ne log:( – )) = 0 in (0,27) is
2: 1313
a. 3/2
c. 9/2
b. 4
d. 13/3
11
799Illustration 4.4 Solve
log(-x-6x)/10 (sin 3x+sin x) = log-x2-6x)/10 (sin 2x)
11
8007. For the smallest positive values of x and y, the equation
2(sin x + siny) – 2 cos(x – y) = 3 has a solution, then
which of the following is/are true?
a. sin *+=
1
b
. cos
c. number of ordered pairs (x, y) is 2
d. number of ordered pairs (x, y) is 3
11
80184. elsin xl + e sin xl + 4a = 0 will have exactly four different
solutions in [0, 21) if
a. a E R
b. a e
c. cele
c. ae –
d
d. none of these
. none of these
11
802Find the cosine of ( 8^{circ} 12 ).
A . 0.84
B. 0.54
c. 0.44
D. 0.98
11
803If Q, B, y, d are the solutions of the equation tan
0+
3 tan 30, no two of which have equal tangents.
4. The value of tan a + tan ß + tan y + tand is
a. 1/3
b. 8/3
c. – 8/3
d. 0
11
804Prove that ( 1+tan 2 theta tan theta=sec 2 theta )11
805( boldsymbol{R}=sqrt{boldsymbol{P}^{2} boldsymbol{Q}^{2}-boldsymbol{2} boldsymbol{P} boldsymbol{Q} sin (boldsymbol{theta}-boldsymbol{9} boldsymbol{0})} )
Here ( boldsymbol{P}=mathbf{1 0 1} boldsymbol{g}, boldsymbol{Q}=mathbf{1 0 6} boldsymbol{g}, ) and ( boldsymbol{theta}= )
( mathbf{1 0 9}^{boldsymbol{O}} )
11
806( f frac{sqrt{1+sin frac{39 pi}{8}}}{sqrt{1+sin frac{57 pi}{8}}}=tan left(frac{k pi}{16}right) ) then
least positive value of ( k ) is
A . 1
B. 3
( c .5 )
D.
11
807In the given figure, we have ( A C perp C D, B C perp C D ) and ( D A=D B )
then ( C A=C B ).
A. True
B. False
11
808The range of ( f(x)=operatorname{cosec}^{2} x+ )
( 25 sec ^{2} x ) is ( (a, infty] ) Find ( a )
11
809The appropriate value is ( cos 61^{circ} ) is
A . 0.4848
B. 0.4849
c. 0.4948
D. 0.5059
11
810The general values of ‘ ( theta ) ‘ satisfying the
equation ( sec 4 theta-sec 2 theta=2 ) is
11
8118. Assume that is a rational multiple of it such that cos A
is a distinct rational. The number of values of cos O is
a. 3
b. 4
c. 5
d. 6
11
812Convert ( 25^{0} ) into Radian measure.11
81369. If cost 0 – sin4 0 = 2, then the
value of 2 cos? 0 – 1 is
(1) 0
(2) 1
11
814The number of solutions of the equation
( |sin x|=|cos 3 x| operatorname{in}[-2 pi, 2 pi] ) is:
A . 32
B . 28
c. 24
D. 30
11
8159. Let A = sin x + cos x. Then find the value of sin* x + cos*x
in terms of A.
11
816e tan x
tan y
= tan 2, x + y + z = n and tanx + tan” y
tan z
5
2
3
+ tan= * then K =
+ tan-z=
then K=
K
11
8179. The value of tan(a + B) is11
818Prove that ( frac{2 tan x}{1+tan ^{2} x}=sin 2 x )11
819Find the value of ( boldsymbol{theta}, ) if
( cot theta=0.2334 )
11
82017.
The number of solutions of the equation sin(e)* = 5x +5*;
(1990 – 2 Marks
(a) 0
(b) 1
© 2
(d) Infinitely many
11
821Illustration 3.2 Let A, B, C be the three angles such that
A + B + C = t. If tan A. tan B = 2, then find the value of
cos A cos B
cos C
11
822If ( a=cos 3 ) and ( b=sin 8 ) then:
( mathbf{A} cdot a>0, b>0 )
B. ( a bb )
D. ( a b>0 )
11
823( left(1+tan ^{2} thetaright) cdot sin ^{2} theta= )
( mathbf{A} cdot sin ^{2} theta )
B ( cdot cos ^{2} theta )
( mathbf{c} cdot tan ^{2} theta )
( mathbf{D} cdot cot ^{2} theta )
11
824If ( A+B+C=180^{circ}, ) then the value of
( (cot B+cot C)(cot C+ )
( cot A)(cot A+cot B) ) will be
( mathbf{A} cdot sec A sec B sec C )
B. ( csc A csc B csc C )
( mathbf{c} cdot tan A tan B tan C )
D.
11
825( ln ) a ( Delta A B C, angle A>angle B . ) If ( sin A ) and
( sin B ) satisfy the equation ( 3 sin x- )
( 4 sin ^{3} x-k=0,0<k<1, ) then ( angle C ) is
A ( cdot frac{pi}{3} )
в. ( frac{pi}{2} )
c. ( frac{2 pi}{3} )
D. ( frac{5 pi}{6} )
11
8266. Ifsin-x – 2 sinx-1=0 has exactly four different solutions
in xe [0, nn), then value/values of n is/are (n e M)
b. 3
c. 4
d. 6
a. 5
11
82798. If o, B, y are acute angles and cos 0 = sin B/sin a, cos
= sin / sin a and cos (0 – 0) = sin ß sin y, then the value
of tan’a – tanºß – tany is equal to
a. – 1
b. 0
2 . 1
11
828Find the value of ( 5 sin 30^{0}+3 tan 45^{0} )11
829If ( tan alpha ) and ( tan beta ) are the roots of the
equation ( boldsymbol{x}^{2}+boldsymbol{p} boldsymbol{x}+boldsymbol{q}=boldsymbol{0}(boldsymbol{p} neq boldsymbol{0}), ) then
This question has multiple correct options
A ( cdot sin ^{2}(alpha+beta)+p sin (alpha+beta) cos (alpha+beta)+q cos ^{2}(alpha+beta)= )
( q )
B. ( tan (alpha+beta)=frac{p}{q-1} )
( mathbf{c} cdot cos (alpha+beta)=1-q )
( mathbf{D} cdot sin (alpha+beta)=-p )
11
830Express in Degrees:
( (a)left(frac{2 pi}{15}right)^{c} )
( (b)(-2)^{c} )
A ( cdot(a) 14^{circ} )
( (b) 244^{circ} 32^{prime} 44^{prime prime} )
B . ( (a) 74^{circ} )
( (b)-114^{circ} 32^{prime} 44^{prime prime} )
c. ( (a) 14^{circ} )
( (b)-120^{circ} 32^{prime} 44^{prime prime} )
D cdot ( (a) 24^{circ} )
( (b)-114^{circ} 32^{prime} 44^{prime prime} )
11
831The value of ( sin frac{theta}{2} cdot sin frac{7 theta}{2}+ )
( sin frac{3 theta}{2} cdot sin frac{11 theta}{2}-sin 2 theta cdot sin 5 theta ) is equal
to
( mathbf{A} cdot mathbf{0} )
B.
( mathbf{c} cdot sin 6 theta-cos 5 theta )
( D cdot sin 6 theta+sin 7 theta )
11
83220. A right angle is divided into three positive parts a, B and
y. Prove that for all possible divisions tana + tan ß +
tan y> 1 + tan a tan ß tan y.
11
833Illustration 4.14 If the equation a sinx + cos2x = 2a – 7
possesses a solution, then find the values of a.in
11
834If ( boldsymbol{m} tan left(boldsymbol{theta}-mathbf{3 0}^{circ}right)=boldsymbol{n} tan left(boldsymbol{theta}+mathbf{1 2 0}^{circ}right) )
then ( frac{m-n}{m+n} ) is equal to
A ( .2 cos 2 theta )
B. ( 2 sin ^{2} theta )
c. ( 1 /(8 cos 2 theta) )
D. ( 1 /(8 sin 2 theta) )
11
83549. If tan²0=2 tanềo + 1, then cos 20+ sin-o equals
a. -1
b. 0
c. 1
d. none of these
11
836then –
2 sin e
15. If x=
1+cos + sino’
a. 1 + x
c. x
1-cos + sin e
– is equal to
1+ sin o
b. 1-X
d. 1/8
11
837Illustration 4.16
Solve 2 cos²0+ 3 sin 0 = 0.
11
838f ( tan theta+sin theta=m ) and ( tan theta-sin theta= )
( n, ) then prove that ( m^{2}-n^{2}=4 sqrt{m n} )
11
83919. IfX=sin ( 0 +7.) + sin( 0 – 6 )+ sin( 0 + 32)
Y=cos 0 +) = cos (6-7) + cos( 0 + ?
Y
3 X
then prove that —
Y
= 2 tan 20.
X
11
840Evaluate each of the following in the
simplest form:
( cos 60^{circ} cos 30^{circ}-sin 60^{circ} sin 30^{circ} )
11
841Illustration 3.59 Prove that
Vsinx+4 cos²x – cos*x+4 sin²x = cos 2x.
11
842If u = Vacose + b2 sine + Vasino + b2 cos?
then the difference between the maximum and minimum
values of – is given by
12004)
(a) (a – b)2
(b) 2a + b2
(c) (a+b)
(d) 2(a? +62)
11
843Value of ( tan 15^{circ} ) is
This question has multiple correct options
A ( cdot frac{sqrt{3}-1}{sqrt{3}+1} )
B. ( 2-sqrt{3} )
( c cdot 2+sqrt{3} )
D. ( sqrt{3}-1 )
11
844The value of ( frac{6 pi^{c}}{5} ) in sexagesimal
measure is
A . ( 144^{circ} )
B . ( 216^{circ} )
( c cdot 240^{9} )
D. ( 120^{9} )
11
845If ( A+B+C=pi ) then which of the
following are true?
i. ( tan 3 A+tan 3 B+tan 3 C= )
( tan 3 A tan 3 B tan 3 C )
ii. ( cot frac{A}{2}+cot frac{B}{2}+cot frac{C}{2}= )
( cot frac{A}{2} cot frac{B}{2} cot frac{C}{2} )
A. only 1st statement is true
B. only 2nd statement is true
c. both statements are true
D. both statements are false
11
846If sin a = A sin(a+ß), A #0, then
1. The value of tan a is
Asin B
Asin ß
a.
b. –
1- Acos B
1+ Acos ß
A cos
B
d .
Asin ß
1 – Asin
B
1 + Acos ß
11
847Solve:
( left(sqrt{3}+tan 1^{0}right) )
11
848Solve ( tan theta+tan 2 theta+ )
( sqrt{3} tan theta tan 2 theta=sqrt{3} )
A. ( theta=frac{n pi}{3}+frac{pi}{3}, n in Z )
в. ( theta=frac{n pi}{3}+frac{pi}{6}, n epsilon Z )
c. ( theta=frac{n pi}{3}+frac{pi}{12}, n in Z )
D. ( theta=frac{n pi}{3}+frac{pi}{9}, n in Z )
11
849Illustration 3.98 If x, y e R and x2 + y2 + xy = 1, then find
the minimum value of xy + xy + 4.
11
850The value of ( frac{4}{tan ^{2} 60^{0}}+frac{1}{cos ^{2} 30^{0}}- )
( sin ^{2} 90^{0} ) is equal to
11
851Illustration 3.67 Find the value of cos 12° + cos 84° +
cos 156° + cos 132º.
11
8525.
The general solution of the trigonometric equation sin x+cos
x=1 is given by :
(1981 – 2 Marks)
(a) x = 2nt ; n=0, #1, #2…
(b) x = 2n1 + n/2;n=0, +1, 2…
(C) x=nn+(-1)”
(d) none of these n=0, +1, +2…
11
853Illustration 3.15
Prove that –
cos 10° + sin 10°
-= tan 55º.
cos 10º – sin 10°
11
854Assertion ( f(x)=frac{2}{pi} x sin x+x^{3}, ) where ( x in )
( left[0, frac{pi}{2}right] )
Statement-1: ( f(x)=frac{pi}{2} ) has exactly one solution in ( boldsymbol{x} inleft[mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right] )
and
Reason
Statement-2: ( boldsymbol{f}(boldsymbol{x}) geq mathbf{0} ) for all ( boldsymbol{x} ) in
( left[0, frac{pi}{2}right] )
A. Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1
c. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.
11
855Illustration 3.28 If sin A = sin B and cos A = cos B, then
A B a
prove that sin = 0.
11
856Which one of the following is not correct
( ? )
A ( cdot frac{(sin A+sin B)}{(sin A-sin B)}=frac{tan frac{1+beta}{2}}{tan frac{1-B}{2}} )
( mathbf{B} cdot sin ^{2} A-cos ^{2} B=sin (A+B) sin (A-B) )
C ( cdot cos A-cos B=2 cos frac{A+B}{2} cos frac{B-A}{2} )
( mathbf{D} cdot cos ^{2} A-cos ^{2} B=sin (A+B) sin (B-A) )
11
857Illustration 4.26
Solve sec 40 – sec 20= 2.
11
858Illustration 4.8 If sin A = sin B and cos A = cos B, then find
the value of A in terms of B.
11
859For any ( theta ), state the value of:
( sin ^{2} theta+cos ^{2} theta )
11
860Illustration 4.41 Solve logtan (2 + 4 cos²x) = 2.11
861If ( A, B ) are acute angles, ( sin A= ) ( frac{4}{5}, tan B=frac{5}{12} ) then ( sin (A+B)= )
A ( cdot frac{36}{65} )
в. ( frac{65}{56} )
c. ( frac{65}{63} )
D. ( frac{63}{65} )
11
862Suppose ( x ) and ( y ) are real numbers such that ( tan x+tan y=42 ) and ( cot x+ )
( cot y=49, ) then the single digit prime
number by which the value of
( tan (x+y) ) is not divisible is
11
863If ( frac{x}{y}=frac{cos A}{cos B} operatorname{then} frac{x tan A+y tan B}{x+y}= )
A. ( cot frac{A+B}{2} )
B. ( cot frac{A-B}{2} )
c. ( tan frac{A-B}{2} )
D. ( tan frac{A+B}{2} )
11
86414. If 2tan´x – 5sec x = 1 is satisfied by exactly seven
distinct values of xe |,ne N, then the
2
greatest value of n is
11
865Simplify:
( frac{1-cos x}{1+cos x} )
11
86664. If A is an acute angle and cot A+
cosec A = 3, then the value of
sin A is
(1) 1
(2)
(4) O
11
867The number of solutions of the equation ( |cot x|=cot x+frac{1}{sin x}(0 leq x leq 2 pi) ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
( D )
11
868Prove that:
[
cos 20^{circ}+cos 100^{circ}+cos 140^{circ}=0
]
11
86953. One of the general solutions of 13 cos 0 – 3 sin 0 = 4
sin 20 cos 30 is
a. mn + /18, me Z b. mtt/2 + Tt/6, V me Z
c. m m/3 + 7/18, me Z d. none of these
11
870Prove that: ( frac{sin A-2 sin ^{3} A}{2 cos ^{3} A-cos A}=tan A )11
871Illustration 3.81 If A
tan A
tan B
tan B.tan C tan A. tan C
– 2cotA – 2cot B – 2cotC
+ B + C = 1, prove that
tan C
=tan A + tan B + tan C
tan A. tan B
11
87249. The set of values of x satisfying the equation sin 3a = 4
sin a sin(x + a) sin(x – a) is
a. nn + Tt/4, v nez b. nnt + 7/3, Vnez
c. na = 7/9, Vnez d. ntt + Tt/12, Vnez
11
8736. The general values of 0 satisfying the equation 2 sin’e-3
sin 0-2 = 0 is (n e Z)
a. nt+ (-1)” 7/6 b. TTC+(- 1)” T/2
c. nt+ (-1)” 570/6 d. nnt + (-1)” 770/6
11
87420. 272 – 1 is equal to
a. sin a
c. sin
b. cos a
d. cos e
11
875Prove that:
( frac{sin A-2 sin ^{3} A}{2 cos ^{3} A-cos A}=tan A )
11
876Illustration 4.2
Solve
tan 3x – tan 2x
1 + tan 3x tan 2x
= 1.
11
877If ( sin alpha+operatorname{cosec} alpha=2, ) find the value of
( sin ^{n} alpha+operatorname{cosec}^{n} alpha, n epsilon Z )
11
878Prove that ( frac{sin 60^{circ}+sin 30^{circ}}{sin 60^{circ}-sin 30^{circ}}= )
( frac{tan 60^{circ}+tan 45^{circ}}{tan 60^{circ}-tan 45^{circ}} )
11
879Find the acute angles ( A ) and ( B ) satisfying ( sec A cot B-sec A-2 cot B+2=0 )11
880Illustration 3.1 Prove that
sin (B-C) sin(C – A)
cos B cos C cos Ccos A
sin(A-B)
cos A cos B = 0.
11
881( f-1+cos 56^{circ}+cos 58^{circ}+cos 66^{circ}= )
( k sin 28^{circ} sin 29^{circ} sin 33^{circ}, ) then the value
of k is
A . 1
B. 3
( c cdot 4 )
D. 5
11
882Illustration 3.80 If A + B + C = Tt, prove that cot A + cot B
+ cot C – cosec A . cosec B. cosec C = cot A . cot B. cot C.
11
883If ( tan (A+B)=1, ) and ( cos (A-B)= )
( frac{sqrt{3}}{2}, 0^{o}<A+BB ; ) find
( A ) and ( B )
11
884Illustration 3.75 If A + B + C = 180°, prove that cos? A +
cos B + cos-C=1 – 2 cos A cos B cos C.
11
885cot 25° + cot 55°
Illustration 3.20 Find the value of –
tan 25° + tan 550 +
cot 55° + cot100°
tan 55° + tan 100°
cot100° + cot 25°
tan 100° + tan 25°
11
886( sin 85^{circ}-sin 35^{circ}-cos 65^{circ}= )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
11
887The value of ( cos ^{2} 30^{0}-cos ^{2} 60^{0}- )
( cos 60^{0} ) is
( mathbf{A} cdot mathbf{0} )
в. ( frac{1}{2} )
( c cdot frac{3}{4} )
D.
11
888The number of solutions of ( 3 sec theta-5= )
( 4 tan theta operatorname{in}[0,4 pi] ) be ( k . ) Find ( k ? )
11
88912-sin a – cos a in
is equal to
sin a – cos a
α
π
a. sec
b. cos
Bloo
o
2
a
c. tan
Bloo
d. cot
11
890Find the general solution of ( sec theta+1= ) ( (2+sqrt{3}) tan theta )11
891The set of values of a for which the
equation ( sin x(sin x+cos x)=a ) has
real solutions is
A ( cdot[1-sqrt{2}, 1+sqrt{2}] )
B. ( [2-sqrt{3}, 2+sqrt{3} )
D. ( left[frac{1-sqrt{2}}{2}, frac{1+sqrt{2}}{2}right] )
11
892The value of ( cos ^{4}left(frac{pi}{4}right)-cos ^{4}left(frac{pi}{6}right)+ )
( sin ^{4}left(frac{pi}{6}right)+sin ^{4}left(frac{pi}{3}right) ) is
A ( cdot frac{1}{16} )
в. ( frac{1}{8} )
c. ( frac{5}{16} )
D. ( frac{3}{16} )
11
893Find the general solution:
( sin x+sin 3 x+sin 5 x=0 )
11
8945. For all in [0, r/2] show that cos(sin O) sin(cos 6).
CITLIFE 1981
11
895The value of ‘ ( b^{prime} ) such that the equation ( frac{b cos x}{2 cos 2 x-1}=frac{b+sin x}{left(cos ^{2} x-3 sin ^{2} xright) tan x} )
possess solution, then prove that belongs to the set ( left(-infty, frac{1}{2}right) )
11
896If ( frac{sin ^{3} theta-cos ^{3} theta}{sin theta-cos theta}-frac{cos theta}{sqrt{1+cot ^{2} theta}}- )
( 2 tan theta cot theta=-1, theta in[0,2 pi], ) then
A ( cdot theta inleft(0, frac{pi}{2}right)-left{frac{pi}{4}right} )
в. ( _{theta} inleft(frac{pi}{2}, piright)-left{frac{3 pi}{4}right} )
c. ( _{theta inleft(pi, frac{3 pi}{2}right)-left{frac{5 pi}{4}right}} )
D. ( theta in(0, pi)-left{frac{pi}{4}, frac{pi}{2}right} )
11
897if ( boldsymbol{x}, boldsymbol{y} boldsymbol{epsilon}[mathbf{0}, boldsymbol{2} boldsymbol{pi}] ) and ( sin boldsymbol{x}+sin boldsymbol{y}=boldsymbol{2} ) then
the value of ( x+y ) is
A . ( pi )
в.
( c .3 pi )
D. None of these
11
898( left{mathbf{x} in mathbb{R}: cos 2 x+2 cos ^{2} x-2=0right}= )
A ( cdotleft{2 n pi+frac{pi}{3}, n in Zright} )
в ( cdotleft{n pi pm frac{pi}{6}, n in Zright} )
с ( cdotleft{n pi+frac{pi}{3}, n in Zright} )
D ( cdotleft{2 n pi-frac{pi}{3}, n in Zright} )
11
899Illustration 3.6 If 3tan 8 tan o = 1, then prove that
2 cos(0+ 0) = cos(0-0).
11
900Statement
(I): If
( sin alpha=frac{12}{13},left(0<alpha<frac{pi}{2}right) ) and
( cos beta=-frac{3}{5},left(pi<beta<frac{3 pi}{2}right) ) then
( sin (alpha+beta)=frac{56}{65} )
Statement (II): If
( theta ) and ( phi ) are angles in the first quardrant such that ( tan theta=frac{1}{7} ) and ( sin phi=frac{1}{sqrt{10}} operatorname{then} theta+2 phi=45^{circ} )
Which of the above statements is
correct?
A. only
B. only ॥
c. Both I&॥
D. Neither I nor I
11
901If ( tan alpha=K cot beta, ) then ( frac{cos (alpha-beta)}{cos (alpha-beta)} )
equals
A ( cdot frac{1+K}{1-K} )
в. ( frac{1-K}{1+K} )
c. ( frac{K+1}{K-1} )
D. ( frac{K-1}{K+1} )
11
902Illustration 3.5 Show that cos 0 + cos²(a + ) – 2 cos a
cos e cos(a + 0) is independent of e.
11
903Find the value of ( sin 765^{circ} )11
904if ( alpha ) and ( beta ) are angles in the first quadrant, ( tan alpha=frac{1}{7}, sin beta=frac{1}{sqrt{10}}, ) then
using the formula ( sin (A+B)= )
( sin A cos B+cos A sin B, ) one can find
the value of ( (boldsymbol{alpha}+mathbf{2} boldsymbol{beta}) ) to be
( A cdot 0^{circ} )
B. 45
( c cdot 60 )
( D cdot 90^{circ} )
11
905
23. The expression cos 30 + sin 30 + (2 sin 20 – 3) (sin
cos ) is positive for all in
a. (ann – 31 , 2n + 7), nez
b. (2nt – , 2nn + ) nez
4
11
906if ( sec theta+tan theta=x, ) then what is the
value of ( sec theta ? )
11
90710. Show that the value of , wherever defined, never
tan 3x
lies between – and 3.
(IIT-JEE 1992)
11
908( f cos ^{2} 2 x-cos ^{2} 6 x=sin m x sin 8 x )
Find ( boldsymbol{m} )
11
9091. Prove that
sin x – cos x +1
sin x + cos x -1
= secx + tan x.
11
910Illustration 2.20
Express 1.2 rad in degree measure.
11
911( cos 20^{circ} cos 100^{circ}+cos 100^{circ} cos 140^{circ}- )
( cos 140^{circ} cdot cos 200^{circ} ) is equal to
( A cdot frac{3}{4} )
B. ( frac{1}{4} )
( c cdot-frac{1}{4} )
D. ( -frac{3}{4} )
11
912If ( frac{tan 2 theta+tan theta}{1-tan theta tan 2 theta}=0, ) then the
general value of ( boldsymbol{theta} ) is
A ( , n pi ; in I )
( I )
B. ( frac{n pi}{3} ; in I )
c. ( frac{n pi}{4} ; in I )
D. ( frac{n pi}{6} ; in I )
11
913( sin 360^{circ}=? )11
9146. Let e, OE 10, 27t] be such
that 2 cos 0 (1 – sin
) =
sin e tan
+cot
cos 0 – 1, tan(21 – 0) > 0 and
-1<sin
. Then q cannot satisfy
(IIT-JEE 2012)
11
915( frac{sin (alpha+beta)}{sin (alpha-beta)}=frac{a+b}{a-b} ) then prove that
( boldsymbol{a} tan boldsymbol{beta}=boldsymbol{b} tan boldsymbol{alpha} )
11
916Given ( tan (pi cos theta)=cot (pi sin theta) )
then the value of ( cos left(theta-frac{1}{4} piright) ) will be
( ^{A} cdot frac{1}{2 sqrt{2}} )
B. ( frac{1}{sqrt{2}} )
c. ( frac{1}{3 sqrt{2}} )
D. ( frac{1}{4 sqrt{2}} )
11
917Illustration 4.58
Find the number of solutions of sinx= –
11
918If ( tan A=frac{1}{2}, tan B=frac{1}{3} ) then
( tan (A+B)= )
A .
в.
c. -1
D.
11
9191.
m
and tanſ =
find the possible values
If tana=-
m+1
of(a+b).
2m +1
(1978)
11
92021. Which of the following set of values of x satisfies the
equation 2(2 sin x-3 sin x+1) + 2(2-2 sin? x +3 sin x) = 9?
TT
a. X=n7+
,ne I
b. x=nnt
,ne I
c. x=nt, ne
I
d
. x= 2n1+
,ne I
11
92160. If 4 sin 20+ sin²0 = 4, then what
will be the value of tan (90° + 0)
from the following ?
(1) O
11
92213. Show that the equation esinx-e-sin x – 4 = 0 has no real
solution.
(1982 – 2 Marks)
11
9231. Number of values of p for which equation sin’x + 1 +
p3 – 3 p sinx = 0 (p > 0) has a root is
11
924Find the value of
( cos 1^{0} cos 2^{0} cos 3^{0} ldots . . cos 89^{0} ldots . . cos 179 )
11
925Solve the following equations.
( cos 9 x-2 cos 6 x=2 )
11
926If ( alpha ) is only real root of the equation
( boldsymbol{x}^{3}-(cos 1) boldsymbol{x}^{2}+(sin 1) boldsymbol{x}+1=mathbf{0}, ) then
( left(tan ^{-1} alpha+tan ^{-1} frac{1}{alpha}right) ) cannot be equal
to
his question has multiple correct options
( mathbf{A} cdot mathbf{0} )
в. ( frac{pi}{2} )
( c cdot-frac{pi}{2} )
D.
11
927Prove that ( cos ^{2} 45^{circ}-sin ^{2} 15^{circ}=frac{sqrt{3}}{4} )11
928The number of solutions of the equation ( 1+sin ^{4} x=cos ^{2} 3 x, x inleft[-frac{5 pi}{2}, frac{5 pi}{2}right] )
is?
( mathbf{A} cdot mathbf{5} )
B. 4
( c cdot 7 )
( D )
11
929( sin left(frac{pi}{2}-xright)=cos x )11
93022. In a right angled triangle, acute angles A and B satisfy
tan A + tan B + tan-A + tan-B + tan A + tanB = 70.
Find the angle A and B in radians.
11
931In an Isosceles triangle ( A B C, tan ^{2} B- )
( sec ^{2} B+2 )
11
932If ( mathrm{B} ) be the exterior angle of a regular
polygon of ( n ) sides and ( A ) is any constant, then prove that ( sin A+sin (A+B)+sin (A+2 B)+ )
( . . n ) terms ( =0 )
11
933The number of solutions of the equation ( frac{sec x}{1-cos x}=frac{1}{1-cos x} ) in ( [0,2 pi] ) is
equal to
( A cdot 3 )
B. 2
( c )
( D )
11
934Illustration 4.13 Find the number of solutions of the
equation esinx – e-sinx – 4 =0.
11
935Prove that ( : frac{1}{sin 10^{circ}}-frac{sqrt{3}}{cos 10^{circ}}=4 )11
936In a circle of diameter ( 40 mathrm{cm} ), the length
of a chord is ( 20 mathrm{cm} . ) Find the length of minor arc of the chord.
11
937c
10. Let cos(a+b) = andsin (a–B) = 13 where
osa,Bs.Then tan 2a =
[2010]
11
938then
17. If y = (1 + tan A) (1 – tan B), where A – B
(y + 1)p+l is equal to
zista.9
b. 4
c. 27
d. 81
11
939The value of expression ( frac{(2 tan 4+3 cot 4)(2 cot 4+3 tan 4)}{24 cot ^{2} 8+25} ) is
( A )
B. 2
( c cdot 3 )
( D )
11
940If ( sin theta=-frac{4}{5}, pi<theta<frac{3 pi}{2}, ) then find
i) ( sin 2 theta )
ii) ( cos 2 theta )
iii) ( tan 2 theta )
11
941What is the ( sin (boldsymbol{alpha}+boldsymbol{beta})- )
( 2 sin alpha cos beta+sin (alpha-beta) ) equal to?
( mathbf{A} cdot mathbf{0} )
B. ( 2 sin alpha )
( c cdot 2 sin beta )
( D cdot sin alpha+sin beta )
11
942Illustration 4.18
Solve sin
cos
– cos O sin 0= 1/4.
11
943( sec ^{2} 50^{circ}-cot ^{2} 40^{circ}-sin ^{2} 45^{circ}=? )
A ( cdot 1 / 2 )
B. 5
( c cdot 1 )
D.
11
944Find the radian measures
corresponding to the following degree
measures:
( 25^{circ} )
( -47^{circ} 30 )
( 240^{circ} )
( 520^{circ} )
11
94539
6. Suppose a, and B, are the roots of the
12
equation x2 – 2x sec a + 1 = 0 and a, and B, are the roots of
the equation x2 + 2x tan 0-1=0.Ifa, >, and > B2, then
aj + B2 equals
(JEE Adv. 2016)
(a) 2 (sec -tan )
(b) 2 sec
(c) -2 tano
(d) 0
11
94619. The number of solution of sinºx – cos²x sin x + 2 sin’x +
sin x = 0 in 0 SX S 31 is
a. 3
b.4
d. 6 )
c. 5
11
947Find the range if ( [2 sin x]+[cos x]= )
( -3, ) then the range of the function
( f(x)=sin x+sqrt{3} cos x ) in ( [0,2 pi] )
(where [.] denotes the greatest integer function)
A ( cdot(2,-1) )
B ( cdotleft(-1,-frac{1}{2}right) )
( mathbf{c} cdot(-2,-1) )
D. None of these
11
94835. Let A = sin80+ cos14e; then for all real e
a. A 21
b. 0<ASI
c. -<AS-
d. none of these
11
949Prove that ( frac{sin x-sin 3 x}{sin ^{2} x-cos ^{2} x}=2 sin x )11
950Find the principal and general solutions of the following equation:
( sec x=2 )
A ‘ principal solution ( =frac{pi}{6}, frac{5 pi}{6} ) and General solution ( = )
( n pi pm frac{pi}{6}, n in Z )
B. Principal solution ( =frac{pi}{3}, frac{5 pi}{3} ) and General solution ( = )
( n pi pm frac{pi}{3}, n in Z )
C ‘ principal solution ( =frac{pi}{6}, frac{5 pi}{6} ) and General solution ( = )
( 2 n pi pm frac{pi}{6}, n in Z )
D. principal solution ( =frac{pi}{3}, frac{5 pi}{3} ) and General solution ( = )
( 2 n pi pm frac{pi}{3}, n in Z )
11
951A lamp post stands on a horizontal plane. From a point situated at a distance ( 150 mathrm{m} ) from its foot, the angle
of elevation of the top is ( 30^{circ} . ) What is the
height of the lamp post?
A . ( 50 m )
в. ( 50 sqrt{3} m )
c. ( frac{50}{sqrt{3}} m )
D. ( 100 m )
11
952Prove that ( (operatorname{cosec} theta-cot theta)^{2}=frac{1-cos theta}{1+cos theta} )11
95373. cos’x sin 2x = a, sin(rx) Vxe R, then
x=0
a. n= 5, a, = 1/2 b. n=5, a, = 1/4
c. n=5, a, = 1/8 d. n=5, a, = 1/4
11
95468. Given that (1+11+x) tan y =1+v1- x. Then sin 4y is
equal to
a. 4x
b. 2x
c. X
d. none of these
11
955If ( sin theta=frac{1}{2}, ) show that ( (3 cos theta- )
( left.4 cos ^{3} thetaright)=0 )
11
956Number of value of ( boldsymbol{x} in[mathbf{0}, mathbf{4} boldsymbol{pi}] ) and
satisfying ( sqrt{2} sec x+tan x=1 ) is?
11
957Illustration 2.4
Convert 45° to radians.
11
958Prove that ( y=frac{4 sin theta}{(2+cos theta)}-theta ) is an
ncreasing function of ( boldsymbol{theta} ) in ( left[mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right] )
11
95958. The number of values of 0 satisfying 4 cos 0 + 3 sin 0 =
5 as well as 3 cos + 4 sin = 5 is
a. one
b. two
c. zero
d. none of these
11
9605. If sin 0, + sin 02 + sin 02 = 3, then cos 6, + cos Oz + cos 03
is equal to
a. 3
b. 2
c. 1
do o
11
961R
7. If A, B, C are angles of a triangle, then 2sin
sin – sin A cot – cos A is
100
= (0)
a. independent of A, B, C 100
b. function of A, B
c. function of C
d. none of these
11
962Illustration 2.53 Solve tan x > cotx, where x € [0,21].11
963Illustration 3.68 Prove that
cos 36° cos 72° cos 108° cos 144° = 1/16.
11
964An angle which is equal to ( 360^{circ} ) is called ( _{text {一一一一一一一 }} ) angle.
A. Right
B. Complete
c. Acute
D. obtuse
11
9651. In a AABC, if tan A/2, tan B/2, tan C/2 are in A.P., then
show that cos A, cos B, cos C are in A.P.
11
966f ( cos (alpha+beta) sin (gamma+theta)=cos (alpha- )
( beta) sin (gamma-theta) . ) show that ( tan theta= )
( tan alpha tan beta tan gamma )
11
967The value of ( frac{tan 45^{circ}}{sin 30^{circ}+cos 30^{circ}} ) is:
This question has multiple correct options
A. ( frac{2}{sqrt{3}+1} )
B.
( frac{1+sqrt{3}}{2} )
D. ( sqrt{3}-1 )
11
968Eliminate ( x ) from equations
( sin (a+x)=2 b ) and ( sin (a-x)=2 c )
A ( cdot frac{(b+c)^{2}}{sin ^{2} a}-frac{(b-c)^{2}}{cos ^{2} a}=1 )
B. ( frac{(b+c)^{2}}{sin ^{2} a}+frac{(b+c)^{2}}{cos ^{2} a}=1 )
( ^{C} cdot frac{(b+c)^{2}}{sin ^{2} a}+frac{(b-c)^{2}}{cos ^{2} a}=1 )
D. ( frac{(b+c)^{2}}{sin ^{2} a}+frac{(b-c)^{2}}{cos ^{2} a}=-1 )
11
969If ( boldsymbol{A}+boldsymbol{B}+boldsymbol{C}=boldsymbol{pi}, ) show that ( sin left(frac{boldsymbol{A}}{mathbf{2}}right)+ )
( sin left(frac{B}{2}right)+sin left(frac{C}{2}right)=1+ )
( 4 sin frac{pi-A}{4} sin frac{pi-B}{4} sin frac{pi-C}{4} )
11
97065. The sum of all the solutions in [0, 41] of the equation
tanx + cotx + 1 = cos(x+is
a. 37
c. 77/2
b. /2
d. 411
11
971The value of ( 60^{g} ) in circular measure is
A ( cdot frac{pi^{c}}{10} )
в. ( frac{3 pi^{c}}{10} )
c. ( frac{2 pi^{c}}{5} )
D. ( frac{pi^{c}}{2} )
11
972Illustration 4.64
Solve sin 0+ V3 cos 021, -< OST.
11
973Which of the following statements are
possible; ( a, b, m ) and ( n ) being non-zero
real numbers?
A ( cdot 4 sin ^{2} theta=5 )
B ( cdotleft(a^{2}+b^{2}right) cos theta=2 a )
C. ( left(m^{2}+n^{2}right) csc theta=m^{2}-n^{2} )
D. ( sin theta=2.375 )
11
974Assertion
If ( tan A+tan B+tan C=3 sqrt{3}, ) then
triangle is equilateral
Reason
( ln Delta A B C, tan A+tan B+tan C= )
( tan A tan B tan C )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion false but Reason is true
11
97513. The value of cosec10° +cosec50° – cosec 70° is11
9767. Suppose A and B are two angles such that A, B € (0, 1),
and satisfy sin A + sin B = 1 and cos A +cos B = 0. Then
the value of 12 cos 2A + 4 cos 2B is
11
97727. The number of real roots of the equation cosec 0 +
sec 0 – 115 = 0 lying in [0, 1] is
a. 6
b. 8
d. 0
c. 4
11
978Prove that
[
frac{sin (theta+phi)-2 sin theta+sin (theta-phi)}{cos (theta+phi)-2 cos theta+cos (theta-phi)}=
]
( tan theta )
11
979Prove ( left(frac{1+tan ^{2} A}{1+cot ^{2} A}right)=left(frac{1-tan A}{1-cot A}right)^{2} )11
980Illustration 3.9 In a triangle ABC, if sin A sin (B-C)=sin C
sin (A – B), then prove that cot A, cot B, cot C are in A.P..
11
981nt
| 70. If 12 sin 0 – cosec 6 2 1 and Ot **,ne Z, then
COS
a. cos 20 2 1/2
c. cos 20 1/4
d. cos 20 < 1/4
11
982If ( cos alpha+cos beta=0=sin alpha+sin beta )
then ( cos 2 alpha+cos 2 beta ) is equal to
A ( .-2 sin (alpha+beta) )
B. ( -2 cos (alpha+beta) )
( mathbf{c} cdot 2 sin (alpha+beta) )
D. ( 2 cos (alpha+beta) )
11
983If ( 0 leq x leq pi ) and ( 81^{sin ^{2} x}+81^{cos ^{2} x}=30 )
then ( x ) is equal to:
A ( cdot frac{pi}{6} )
в. ( frac{pi}{2} )
c.
D. ( frac{3 pi}{4} )
11
98415. If cos x – sin a cot B sin x = cos a, then the value of
tan (x/2) is
a. –tan (a/2) cot (B/2) b. tan (a/2) tan (B/2)
att bo arc. – cot (0/2) tan (B/2) d. cot (0/2) cot (B/2)
11
985If ( sec theta=sqrt{2} ) and ( theta ) lies in first
quadrant. Find value of ( frac{1+tan theta+operatorname{cosec} theta}{1+cot theta-operatorname{cosec} theta} )
11
986sina, cos aan
cos a and
0.
The sides of a triangle are
V1+sin a cosa for some 0<a Then the greatest
angle of the triangle is
[2004]
(a) 150° (6) 90° (c) 120° (d) 60°
hot the
11
98765. sec 50°. sin 40° +
cos 40°. cosec 50° = ?
(1) 2
(2) O
(3) 1 (4) 15
11
988Which of the following statements are
correct?
This question has multiple correct options
( mathbf{A} cdot sin 1>sin 1^{circ} )
B. ( tan 2tan 2 )
D. ( tan 2<tan 1<0 )
11
989( (operatorname{cosec} A-sin A)(sec A-cos A)= )
( frac{1}{tan A+cot A} )
[Hint: Simplify LHS and RHS separately]
11
9905. In triangle ABC, if sin A cos B = 2 and 3 tan A = tan B,
then cotA is equal to
b. 3
c. 4
a. 2
d. 5
:
11
9915. sin 0 + 73 cos 0 = 6x – x? – 11,0 5 Os 41, x € R, holds
for
a. no values of x and O
b. one value of x and two values of 0
c. two values of x and two values of O
d. two point of values of (x, 0)
11
992Illustration 3.56 If sin a + sin ß= a and cos a + cos ß= b,
prove that tan a – B-+ 4-a? – b?
az + 12
11
99341. The roots of the equation 4×2 – 2 15 x + 1 = 0 are
a. sin 36°, sin 18° b. sin 18°, cos 36°
c. sin 36º, cos 18° odd. cos 18°, cos 36°
11
994Illustration 4.12 If xe (0, 21) and ye (0, 2), then find the
number of distinct ordered pairs (x, y) satisfying the equation
9 cos²x + sec’y – 6cos x – 4 sec y + 5 = 0.
11
995Solve for ( x: sin x+sin 2 x+sin 3 x=3 )
where ( x in(0, pi) )
11
996The value of ( sqrt{2}left(cos 15^{circ}-sin 15^{circ}right) ) is
equal to:
A ( cdot sqrt{3} )
B. ( sqrt{2} )
c. 1
D. 2
E. ( 2 sqrt{3} )
11
997Solve:
( x sin 45^{circ} cdot cos 45^{circ} cdot tan 60^{circ}=tan 45^{circ} )
( cos 60^{circ} )
11
9988.
Which of the following number(s) is/are rational?
(1998 – 2 Marks)
(a) sin 15°
(b) cos 15°
c) sin 15° cos 150
(d) sin 15° cos 75°
11
999Prove ( sin frac{8 pi}{3} cos frac{23 pi}{6}+ )
( cos frac{13 pi}{3} sin frac{35 pi}{6}=frac{1}{2} )
11
1000The solution of ( (s e c boldsymbol{theta}+1)=(2+ ) ( sqrt{3}) tan theta(0<theta<2 pi) ) are
( mathbf{A} cdot pi / 6, pi )
в. ( pi / 3, pi / 4 )
c. ( pi / 6,2 pi / 3 )
D. none of these
11
1001In triangle ( A B C, ) right-angled at ( B ), if
( tan A=frac{1}{sqrt{3}}, ) find the value of:
(i) ( sin A cos C+cos A sin C )
(ii) ( cos A cos C-sin A sin C )
11
1002If ( tan left(45^{circ}+thetaright)=sqrt{3} ) and ( 0<theta<20^{circ} )
then the value of ( theta ) is
A ( .10^{circ} )
B . ( 15^{circ} )
( c cdot 20 )
D. ( 5^{circ} )
11
100315. If – sin 0, cos 0, tan 0 are in G.P., then e is equal to
(ne 2)
a. 2n
+
b. 2nnt I
c. NT +(-1)”
d. nt +
11
1004If ( 3 sin theta+4 cos theta=5, ) then the value of
( 4 sin theta-3 cos theta ) is
( A cdot 0 )
B.
( c .5 )
D. none of these
11
1005Illustration 3.60
= tan 90
Prove that (4 cos29º – 3) (4 cos 27° – 3)
11
100663. If =kk #1) then which of the following is not
tan A
true?
A cos A k-1
sin 3 A 2kl 2
a.
cos3A 2
s in A k-1 mall
cot 3 A 1
==
d. none of these
cot A k
11
10072 sin 20
29. Given that, Ano) = –
cos 20 – cos 4ne, and fO) + f(20) +
sin 20
(30) + … +f(no) = ?
sin sine, then the value of u-2 is
11
1008In a ( triangle A B C, angle A ) is greater than ( angle B ). If
the measures of ( angle A ) and ( angle B ) satisfy the
equation ( 2 tan x-kleft(1+tan ^{2} xright)=0 )
where ( k epsilon(0,1), ) then the measure of the
( angle C ) is
A.
в.
( c cdot frac{5 pi}{12} )
D.
11
1009Solve the following equation:
( sin x+sin ^{2} x+cos ^{2} x=0 )
11
1010Expand ( sin (A+B) )11
1011Change the following degree measures
to radian measure: ( 45^{circ} )
( ^{mathbf{A}} cdot frac{pi}{6} ) radians
B ( cdot frac{pi}{3} ) radians
c. ( frac{pi}{4} ) radians
D ( cdot frac{pi}{2} ) radians
11
1012If ( tan theta, 2 tan theta+2,3 tan theta+3 ) are in
G.P, then the value of ( frac{7-5 cot theta}{9+4 sqrt{sec ^{2} theta-1}} )
is
A ( cdot frac{12}{5} )
в. ( frac{-33}{28} )
c. ( frac{33}{100} )
D. ( frac{12}{13} )
11
101380. In a right angled triangle the hypotenuse is 22 times the
perpendicular drawn from the opposite vertex. Then the
other acute angles of the triangle are
and b. I and
Elt
11
1014Prove that ( sin 60^{circ} . cos 30^{circ}- )
( cos 60^{circ} cdot sin 30^{circ}=sin 30^{circ} )
11
101511. If sin A = sin? B and 2 cos? A = 3 cos? B then the triangle
ABC is
a. right angled
b. obtuse angled
c. isoscelesi
d. equilateral
11
10165. Suppose x and y are real numbers such that tan x + tan y=
42 and cotx + coty=49. Then the prime number by which
the value of tan(x + y) is not divisible by is
11
1017The smallest value of an angle whose
sine is ( -frac{sqrt{mathbf{3}}}{mathbf{2}} ) is
( A cdot 30^{circ} )
B . ( 60^{circ} )
( c cdot 120^{circ} )
D. ( 240^{circ} )
11
1018Illustration 3.79 In any triangle ABC, prove that
sin A cos(B – C) + sin’B cos(C – A) + sinC cos(A – B)
= 3 sin A sin B sin C
11
1019If ( boldsymbol{A}+boldsymbol{B}=frac{boldsymbol{pi}}{boldsymbol{4}} ) then value of ( (boldsymbol{1}+ )
( tan A)(1+tan B)= )
( A cdot 4 )
B.
( c cdot 2 )
D. none of these
11
1020Prove the following identity :
[
frac{1}{sin theta+cos theta}+frac{1}{sin theta-cos theta}=frac{2 sin theta}{1-2 cos ^{2} theta}
]
11
102110. The greatest integer less than or equal to –
1
COS 29001
13 sin 250°
11
1022The values of ( x ) in ( left(0, frac{pi}{2}right) ) satisfying the equation ( sin x cos x=frac{1}{4} ) are
A ( cdot frac{pi}{6}, frac{pi}{12} )
в. ( frac{pi}{12}, frac{5 pi}{12} )
c. ( frac{pi}{8}, frac{3 pi}{8} )
D. ( frac{pi}{8}, frac{pi}{4} )
11
10232. Find all the solution of 4 cos²x sin x – 2 sin²x = 3 sinx.
(IIT-JEE 1983)
11
1024If ( tan x+tan left(x+frac{pi}{3}right)+ )
( tan left(x+frac{2 pi}{3}right)=3, ) then
A ( cdot tan x=1 )
B. ( tan 2 x=1 )
( mathbf{c} cdot tan 3 x=1 )
D. None of these
11
10252. a. Draw the graph of y =
(sin x + cos x) from
x=

(IIT-JEE 1979)
b. If cos(a + b) =
€, sin(a – b)
1
, and a, ß lie
, an
between 0 and 77/4, find tan 2 a.
11
1026If ( I_{n}=int_{0}^{frac{pi}{2}} frac{sin ^{2} n x}{sin ^{2} x} d x, ) then ( I_{1}, I_{2}, I_{2}, cdots )
are in
A. A.P
в. G.
c. н.P.
D. none
11
10273. Number of roots of the equation (3 + cos x)2 = 4 –
2 sin®x, x € [0, 51), are
11
102828. Prove that a triangle ABC is equilateral if and only if
tan A+tan B+tan C=
(1998 -8 Marks)
11
1029Illustration 3.19 If tan’A + tanB + tan’C = 3 tan Atan B.
tan C, then prove that triangle ABC is an equilateral triangle.
11
1030The equation (cos p-1) r4 + (cos p)x+ sin p=0
In the variablex, has real roots. Then p can take any value in
the interval
(1990-2 Marks)
(a) (0,21) (b) (-1,0) ©
(a) (0, 1)
2
2
11
1031At 4: 24 p.m., how many degrees has the hour hand of a clock moved from its
position at noon?
A ( cdot 135^{circ} )
B. 134
( mathrm{c} cdot 133^{circ} )
D. ( 132^{circ} )
11
1032Find the value of ( tan left(frac{1}{2} cos ^{-1} frac{sqrt{5}}{3}right) )11
10334. If sin e- *+ y2 +1
-, then x must be
2x
a. 3d to suono
c. 1
b. -2
d. none of these
11
1034Solve ( 16^{sin ^{2} x}+16^{cos ^{2} x}=10 )11
1035Simplify, using trigonometric tables
( sin 30^{circ} 30^{prime}+cos 40^{circ} 20^{prime} )
11
103630. If tan a is equal to the integral solution of the inequality
4×2 – 16x + 15 < 0 and cos B is equal to the slope of the
bisector of the first quadrant, then sin(a + b) sin(a – b)
is equal to
b.
la viw
11
1037For all ( theta ) in ( left[0, frac{pi}{2}right] ) Prove that
( cos (sin theta)>sin (cos theta) )
11
1038If ( boldsymbol{x} cos boldsymbol{theta}=boldsymbol{y} cos left(boldsymbol{theta}+frac{boldsymbol{2} boldsymbol{pi}}{boldsymbol{3}}right)= )
( z cos left(theta+frac{4 pi}{3}right), ) then the value of ( x y+ )
( boldsymbol{y} boldsymbol{z}+boldsymbol{z} boldsymbol{x}= )
( A )
B.
c.
D.
11
103920. The range of y such that the equation in x, y + cos x = sin x
has a real solution is
a. [-2, 2]
b. [- V2, V2]
c. [-1, 1]
d. [-1/2, 1/2]
11
1040Assertion
The equation ( sin x=1, ) has infinite
number of solutions
Reason
The domain of ( f(x)=sin x ) is ( (-infty, infty) )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
11
10419. Number of integral values of a for which the equation
cos²x – sin x + a = 0 has roots when xe (0, 7/2) is
11
104225. The value of 2 sin x + tanx is
sin 3x tan 3x
is
11
10438. a and ß are the positive acute angles and satisfying
equations 5 sin 2B = 3 sin 2a and tan B = 3 tan a
simultaneously. Then the value of tan a + tan ß is
11
10440
HONUT CSO
The number of all possible triplets (a, a, a,) such that
a, + a, cos(2x) +azsin-(x)=0 for all x is (1987-2 Marks)
(a) zero (b) one (c) three
(d) infinite (e) none
16
11
10459. The absolute value of the expression tan
1
97
tan —
16
+ tan
137.
– is
16
11
1046Illustration 3.70 Prove that tan. = 14 + 212 – (V2 + 1)11
1047Illustration 2.2
1+sin e
Prove that – = sec 0 + tan e.
V1–sino
11
104812. If x = sec
– tan o and y = cosec 0 + cot 0, then
y +1
y-1
a.
X=

=
y-1
b.
x=
y +1
um
c.
y =
1 + x
1-x
d. xy + x – y + 1 = 0
11
1049( i f angle A=angle B=45^{circ} ) verify that ( , sin )
( (A+B)=sin A cos B+cos A sin B )
11
105056. Find in radians the angle between
the hour hand and the minute
hand of a clock at half past three.
radians
radians
radians
radians
radians
11
1051Find the value of the other five
trigonometric functions for the
following:
( tan x=frac{3}{4}, x ) in quadrant ( I I I )
11
1052The value of
1) ( sin left(4 theta+frac{pi}{2}right) )
2) ( tan theta+sec theta )
11
1053If ( sin alpha+sin beta=a ) and ( cos alpha-cos beta= )
( b, ) then ( tan frac{alpha-beta}{2} ) is equal to
A. ( -frac{a}{b} )
в. ( -frac{b}{a} )
c. ( sqrt{a^{2}-b^{2}} )
D. None of these
11
1054Find the value of
( frac{sqrt{1+sin 2 A}+sqrt{1-sin 2 A}}{sqrt{1+sin 2 A}-sqrt{1-sin 2 A}} ) when
( |tan A|<1 ) and ( |A| ) is acute
( mathbf{A} cdot tan A )
B. ( -tan A )
( c cdot cot A )
D. – cot ( A )
11
1055State whether true or false:
( sin 2 x+2 sin 4 x+sin 6 x= )
( 4 cos ^{2} x sin 4 x )
A. True
B. False
11
10568.
Find all the solution of 4 cosxsin x – 2 sina x = 3 sin x
(1983 – 2 Marks)
11
1057Find the number of values of ( boldsymbol{theta} )
satisfying the equation ( sin 3 theta= ) ( 4 sin theta cdot sin 2 theta cdot sin 4 theta ) in ( 0 leq theta leq 2 pi )
11
1058If ( 4 cos ^{2} theta+15 cos theta-4=0, ) find the
value of ( left(log _{2} cos thetaright)^{-5}: )
A . -1
в. ( -frac{1}{32} )
( c . )
D. none of these
11
1059Evaluate
( sin 1^{0} sin 2^{0} sin 3^{0} ldots sin 179^{0} sin 180^{0} )
11
1060Illustration 4.19 Solve V5 – 2 sin x = 6 sin x-111
1061Solve ( sin ^{2} theta-cos theta=frac{1}{4} .0 leq theta leq 2 pi )11
106275. If sin* 0 + cos* 0 = 2 sina o
cos2 O, O is an acute angle, then
the value of tan 0 is
(2) 2
(3) 2 (4) O
(1) 1
11
1063If ( tan theta=frac{-4}{3} ) then ( sin theta ) is
A ( cdot frac{-4}{5} ) but not ( frac{-4}{3} )
B. ( frac{-4}{5} ) or ( frac{4}{5} )
c. ( frac{4}{3} ) but not ( frac{-4}{5} )
D. None of thses.
11
1064Area of circle in which a chord of length
( 2 sqrt{3} ) units, subtends angle ( 120^{circ} ) at its
center is-
A . ( pi ) sq units
B. 2 ( pi ) sq units
c. ( 4 pi ) sq units
D. None of these
11
1065If ( sec A+tan A=m ) and ( sec A- )
( tan A=n, ) find the value of ( sqrt{m n} )
( A cdot 0 )
B. ±1
( c .pm 2 )
( D ldots 3 )
11
1066If ( sin x+sin ^{2} x=1 ) then the value of
( cos ^{2} x+cos ^{4} x ) is equal to
( A )
B. ( frac{1}{2} )
c. ( frac{1}{3 sqrt{3}} )
D. ( frac{3 sqrt{5}-5}{2} )
11
1067Illustration 3.69 Show that
4 sin 27° = (5+15)1/2 – (3 – 75)12.
11
1068Prove ( operatorname{cosec}^{6} boldsymbol{A}-cos ^{6} boldsymbol{A}= )
( mathbf{3} cot ^{2} boldsymbol{A} operatorname{cosec}^{2} boldsymbol{A}+mathbf{1} )
11
1069Illustration 3.104 Prove that in a A ABC, sin A + sinB +
sin
sin? Csı
11
1070Find the values of the following:
( sin 120^{circ} )
11
1071Illustration 4.7 Find general value of Owhich satisfies both
sin 0 = -1/2 and tan 0 = 1/73 simultaneously.
7
11
107219. If sin’x cos 3x + cos’x sin 3x = 3/8, then the value of
8sin 4x is
19. If sin x cos 3x + cos’r sin 3x = 318, then the value of
11
107328.
Let S =
x e(-TT,t):
X
0
,-
*. The sum of all distinct
solutions of the equation 13 sec x + cosec x + 2(tan x –
cot x)=0 in the set S is equal to (JEE Adv. 2016)
(b)
276
(©) 0
11
1074Illustration 4.28
Solve cos x cos 2x cos 3x = 1/4.
11
1075If ( tan ^{4} theta+cot ^{4} theta=A, ) then
A ( . A>2 )
в. ( A geq 2 )
c. ( A>4 )
D. ( A geq 4 )
11
1076Solve the following equation:
( 2^{sin ^{2} x}+2^{cos ^{2} x}=2 sqrt{2} )
11
1077Evaluate
( frac{sin ^{2} 63^{circ}+sin ^{2} 27^{circ}}{cos ^{2} 17^{circ}+cos ^{2} 73^{circ}} )
2) ( sin 25^{circ} cos 65^{circ}+cos 25^{circ} sin 65^{circ} )
11
1078If ( sin alpha sin beta-cos alpha cos beta+1=0 ), then
show that ( 1+cot alpha tan beta=0 )
11
1079( 1+cos 56^{circ}+cos 58^{circ}-cos 66^{circ}= )
( m cos 28^{circ} cos 29^{circ} sin 33^{circ} . ) Find ( m )
11
10803. A general solution of tan+ cos20 = 1 is (n € Z)
a. n-
b. 211
+
c. nn +
d. na
11
1081Illustration 3.72
Prove that sin 0 + sin 30 + sin 50 + …
sin?ne
sin e
+ sin(2n – 1)
=
11
1082Find the general solution of the
equation ( sin 2 x+sin 4 x+sin 6 x=0 )
11
108318. The smallest positive value of x (in radians) satisfying the
equation logcosx
SIN X
= 2 – logsec x(tan x), is
11
1084Expand
( cos theta+cos phi= )
11
1085If ( boldsymbol{theta} inleft(mathbf{0}, frac{boldsymbol{pi}}{mathbf{2}}right), ) then the value of
( cos left(theta-frac{pi}{4}right) ) lies in the interval
( ^{mathbf{A}} cdotleft(frac{1}{2}, 1right) )
B ( cdotleft(frac{1}{sqrt{2}}, 1right) )
( ^{mathbf{c}} cdotleft(-frac{1}{sqrt{2}}, 1right) )
D. (0,1)
11
1086Prove that ( cos ^{2} A+cos ^{2}(A+120)+ )
( cos ^{2}(A-120)=frac{3}{2} )
11
1087The total no. of solution of equation ( |cot x|=cot x+frac{1}{sin x}, x in[0,3 pi] ) is
equal to
( A cdot 3 )
B . 2
( c cdot 1 )
( D )
11
108821. In triangle ABC, tan (A – B) + tan (B-C)+ tan (C-A)=0.
Prove that the triangle isisosceles.
11
1089( cot A ) is the product of ( cot ) and ( A )
A. True
B. False
11
1090( boldsymbol{I} boldsymbol{f} sin 5 boldsymbol{theta}=boldsymbol{a} sin ^{5} boldsymbol{theta}+boldsymbol{b} sin ^{3} boldsymbol{theta}+ )
( operatorname{csin} theta ) then
( (A) a-2 c=5 )
(B) ( a-2 c=6 )
(C) ( b+3 c=5 )
(D) ( b+3 c=-5 )
The general solution of ( tan 3 theta tan theta= )
1 is given ( (n in boldsymbol{I}) )
( mathbf{A} cdot a=20, b=-10, c=5 )
B . ( a=1, b=-20, c=5 )
( mathbf{c} . a=16, b=-2, c=5 )
D. ( a=16, b=-20, c=5 )
11
1091Find the measure of an angle in degrees formed by an arc of ( 2.5 mathrm{cm} ) length at the centre of a circle with ( 15 mathrm{cm} ) radius11
1092General solution for ( |sin x|=cos x ) is
A ( cdot 2 n pi+frac{pi}{4}, n in I )
В ( cdot 2 n pi pm frac{pi}{4}, n in I )
c. ( n pi+frac{pi}{4}, n in I )
D. None of these
11
1093Given that ( A ) is positive acute angle and ( sin A=frac{sqrt{3}-1}{2}, ) then ( A ) take the value
( (s)- )
A .15
B. 30
c. 45
D. 75
11
10943. The general solution of the equation sin x – 3 sin 2x +
sin 3x = cos x – 3 cos 2x + cos 3x is (ne 2)
a. nt+
ь пли
та
28
C. (+1)» MT
d. 2nd+ cos!?
11
1095General value of ( theta ) satisfying equation
( tan ^{2} theta+sec 2 theta=1 ) is
A ( . n pi )
B . ( n pi+frac{pi}{3} )
( mathbf{c} cdot n pi-frac{pi}{3} )
D. All of these
11
109619. The maximum value of (cosa,).(COS C.,)…(cosa), under
the restrictions
05@,, ,, …,2., and (cot (,).(cot ay)… (cot Qn)= 1 is
(20015)
(a) 1/22 (6) 1/2 © 1/2n di
11
109712. Let k be sum of all x in the interval [0, 21) such that
3cot-x + 8cot x + 3 = 0, then the value of k/a is
11
1098If
( [x] ) denotes the greatest integer ( leq x )
then the system of linear equations ( [sin theta] x+[-cos theta] y=0[cot theta] x+y=0 )
A ( cdot ) Have infinitely many solutions if ( theta epsilonleft(frac{pi}{2}, frac{2 pi}{3}right) cup ) ( left(pi, frac{7 pi}{6}right) )
B. Have infinitely many solutions if ( theta epsilonleft(frac{pi}{2}, frac{2 pi}{3}right) ) and has unique solution if ( theta epsilonleft(pi, frac{7 pi}{6}right) )
c. Has a unique solution if ( theta epsilonleft(frac{pi}{2}, frac{2 pi}{3}right) ) and have nfinitely many solutions if ( theta epsilonleft(pi, frac{7 pi}{6}right) )
D. Has a unique solution if ( theta epsilonleft(frac{pi}{2}, frac{2 pi}{3}right) cupleft(pi, frac{7 pi}{6}right) )
11
109914. If (1 + tan a) (1 + tan 4a) = 2, a E (0, 1/16), then a is
..
mequal to
it to usd
11
1100Illustration 3.62
If x + y + z = xyz, prove that
2x
I 2x
2y
2z.
1-r2
1- x² 1 – 2 1 – 2²
परx
22 x
11
1101Express the sexagesimal measure ( 15^{circ} )
as radian measure
11
1102(sinx + cosx) from x-
2.
(a) Draw the graph of y=
(6) If cos (a + b) = 5, sin (Q – B) = g, and a, lies
between 0 and -, find tan2a.
(1979)
11
1103The values of ( x epsilon[-2 pi, 2 pi] ) such that
( frac{sin x+i cos x}{1+i}, i=sqrt{-1}, ) is purely
imaginary, are given by
A ( cdot n pi-frac{pi}{4} )
в. ( n pi+frac{pi}{4} )
( c cdot n pi )
D. none of these
11
110424.
The values of 0 € (0,210) for which 2 sin20-5 sino +2>0
are
(2006 – 3M, -1)
48
11
110591. In triangle ABC, tan A + tan B + tan C = 6 and tan A tan B
= 2, then the values of tan A, tan B, tan C are, respectively
soba. 1, 2, 3 tab. 3, 2/3, 7/3
c. 4, 1/2, 3/2
d. none of these
11
1106The range of ( boldsymbol{f}(boldsymbol{x})= )
( frac{1}{5 sin x-6} epsilon[-a,-1 / b] ) Find ( a+b )
11
11072. If A = sin 45° + cos 45º and B = sin 44° + cos 44°, then
a. A > B
b. A<B
c. A=B
d. none of these
11
1108The number of solutions of ( cos x= )
( |1+sin x|, 0 leq x leq 3 pi, ) is
A. 3
B. 2
( c cdot 4 )
D. none of these
11
110925. For 0 < x, y < t, the number of ordered pairs (x, y)
satisfying the system of equations cot?(x – y) –
(1+73)cot(x – y) + V3 = 0 and cos y =
a. 0
c. 2
d. 3
b. 1 b
otto
11
111067. If sec 0 = cosec o, where and
are acute angles, then the value
of cosec (0 + 0) is
(1) 1
(2) O
(3) undefined (4) 12
11
1111Let ( x=sin 1^{circ}, ) then the value of the
( operatorname{expression} frac{1}{cos 0^{circ} cdot cos 1^{circ}}+ )
( frac{1}{cos 1^{circ} cdot cos 2^{circ}}+frac{1}{cos 2^{circ} cdot cos 3^{circ}}+dots+ )
( frac{1}{cos 44^{circ} cdot cos 45^{circ}} ) is equal to
A . ( x )
в. ( frac{1}{x} )
( c cdot frac{sqrt{2}}{x} )
D. ( frac{x}{sqrt{2}} )
11
1112A flag-staff 20 metres long standing on a wall 10 metres high subtends an
angle whose tangent is 0.5 at a point on
the ground. If ( theta ) is the angle subtended
by wall at that point then ( tan theta= )
A . 1
B. ( frac{1}{3} )
c. ( _{1 text { or }} frac{1}{3} )
D.
11
1113U
IL .
Illustration 4.53 Solve sin’x + cos²y= 2sec z for x, y, and z.
11
1114tanº e
11. If sec o
+

7, then prove that |b|slal.
b
a+b
11
1115Illustration 4.50 Find the number of solutions of sin’xcos²x
= 1 + cos² x sinºx in the interval [0, 211].
11
1116Simplify the following expression:
( frac{1+sin +2 x}{(sin x+cos x)^{2}} )
11
1117Evaluate ( : frac{operatorname{cosec}^{2} mathbf{6 3}^{mathbf{0}}+tan ^{2} mathbf{2 4}^{mathbf{0}}}{cot ^{2} mathbf{6 6}^{mathbf{0}}+mathbf{s e c}^{mathbf{2}} mathbf{2 7}^{mathbf{0}}}+ )
( frac{mathbf{s i n}^{2} mathbf{6 3}^{mathbf{0}}+cos mathbf{6 3}^{mathbf{0}} sin mathbf{2 7}^{mathbf{0}}+sin mathbf{2 7}^{mathbf{0}} mathbf{s e c 6}}{mathbf{2}left(operatorname{cosec}^{mathbf{2}} mathbf{6 5}^{mathbf{0}}-mathbf{t a n}^{mathbf{2}} mathbf{2 5}^{mathbf{0}}right)} )
11
1118Transform the following expression
( frac{tan alpha+tan beta}{cot alpha+cot beta}+[cos (alpha- )
( boldsymbol{beta}) boldsymbol{s e c}(boldsymbol{alpha}+boldsymbol{beta})+mathbf{1}]^{-1} )
11
1119( cot theta+tan theta=x ) and ( sec theta-cos theta= )
( y ) then ( left(x^{2} yright)^{2 / 3}-left(x y^{2}right)^{2 / 3} )
11
11202. If ABC is a triangle and tan
tan
pla
, tan
are in H.P.
then find the minimum value of cot B/2.
11
11212.
Ifa+B+ y2, then
(1979)
(©) tan + tan + tan — tan tan tanz
tan
(d) None of these
11
1122Illustration 4.37
Solve 2 sinºx — 5 sinx cos x – 8 cos x=-2.
11
112329. The value of
29. The value of
( m
is equal
(k-1)
(-0)-G
sin
+
}.
sin
a ka
– +
A
6
to
(a) 3-13
© 2(73-1)
(JEE Adv. 2016)
(b) 2(3-13)
(d) 2(2-13)
11
112493. If a sin x + b cos(x + 0) + b cos(x – 0) = d, then the
minimum value of cose is equal to
2.
2161
zla
c.
va? – 4
d. none of these
d. none of these
2 dl
11
1125Solve ( cos (sin theta)=sin (cos theta) )11
1126Prove that:-
( sin 20 sin 40 sin 60 sin 80=frac{3}{16} )
11
1127( cos frac{2 pi}{15} cos frac{4 pi}{15} cos frac{8 pi}{15} cos frac{14 pi}{15} )
A . ( 1 / 4 )
в. ( 1 / 8 )
c. ( 1 / 16 )
D. ( 1 / 32 )
11
1128Illustration 4.17
Solve 4 cos 0 – 3 sec 0 = tan 0.
11
1129The general solution of the equation ( sin ^{100} x-cos ^{100} x=1 ) is
A ( cdot 2 n pi+frac{pi}{3}, n epsilon I )
В ( cdot n pi+frac{pi}{2}, n epsilon I )
c. ( n pi+frac{pi}{4}, n epsilon I )
D. ( 2 n pi-frac{pi}{3}, n epsilon I )
11
113063. If 9 sino + 40 cos0 = 41, then
coso will be
es
BA TL
11
113157. If , and O2 are two values lying in [0, 21] for which
tan 0= 2, then tan tanz is equal to
inupa ar a.
0
2 03 O b. -1 209 to Sleva
c. 2
d. 1
11
1132One angle of a triangle is ( frac{2 x}{3} ) grad, another is ( frac{3 x}{2} ) degrees, whilst the third is ( frac{pi x}{75} ) radians. Express all angles in
degress.
A. Hence three angles of the triangle are ( 43^{circ}, 30^{circ}, 30^{circ} )
B. Hence three angles of the triangle are ( 24^{circ}, 60^{circ}, 96^{circ} )
C. Hence three angles of the triangle are ( 74^{circ}, 27^{circ}, 98^{circ} )
D. Hence three angles of the triangle are ( 30^{circ}, 60^{circ}, 90^{circ} )
11
11338. If a + B = /2 and 3+ y=ą, then tan a equals
a. 2 (tan B+ tan b. tanß+ tan y
c. tan ß + 2 tan y d. 2 tan ß+ tan y
11
113451. If sin x + cosec x + tan y+cot y=4 where x and y el 0,
then tán is a root of the equation
a. O2 + 2a + 1 = 0 с. 02 + 2a – 1 = 0
c. 20-2a-1=0 d. o – a- 1 = 0
11
1135If ( tan 2 A=cot (A-18), ) where ( 2 A ) is an
acute angle, then find the value of ( boldsymbol{A} )
11
1136The value of
( frac{tan ^{2} 60^{0}-2 tan ^{2} 45^{0}+sec ^{2} 30^{0}}{3 sin ^{2} 45^{0} sin 90^{0}+cos ^{2} 60^{0} cos ^{3} 0^{0}} )
A ( cdot frac{49}{12} )
B. ( frac{7}{3} )
c. ( frac{14}{9} )
D.
11
1137Illustration 3.70
Prove that tan—
16
11
1138If ( sqrt{3} tan 2 theta+sqrt{3} tan 3 theta+ )
( tan 2 theta tan 3 theta=1, ) then the general
value of ( theta ) is
A ( . quad n pi+frac{pi}{5} )
B ( cdotleft(n+frac{1}{6}right) frac{pi}{5} )
( ^{c} cdotleft(2 n pm frac{1}{6}right) frac{pi}{5} )
D. ( left(n+frac{1}{3}right) frac{pi}{5} )
11
1139tan
17. The number of distinct real roots of the equation
tan — 21x_=-5
x²+x+1
b. 5
c. 6
d. none of these
2
a. 4
11
1140Illustration 4.57
Solve the equation
cos (sin x + V2 cosro)- tan” x + ” tan?x) = 1.
11
1141Illustration 3.76 Prove that in triangle ABC, cos? A + cos²B
-cosC = 1-2 sin A sin B cos C.
11
1142If ( sec theta+tan theta=k, cos theta= )
A. ( frac{k^{2}+1}{2 k} )
в. ( frac{2 k}{k^{2}+1} )
c. ( frac{k}{k^{2}+1} )
D. ( frac{k}{k^{2}-1} )
11
1143The value of ( cos left(36^{circ}-Aright) cos left(36^{circ}+right. )
( A)+cos left(54^{circ}+Aright) cos left(54^{circ}-Aright) ) is?
( mathbf{A} cdot sin 2 A )
B. ( cos 2 A )
( c cdot cos 3 A )
( mathbf{D} cdot sin 3 A )
11
114420. The value of
cos-10° – cos 10° cos50° + cos250° is:
JEEM 2019-9 April (M
(b) 3/4
(a
©
+ cos200
(1 + cos20°)
(d) 312
11
1145Ma Tuusu-
2. If 2 cos x + sin x = 1, then find the value of 7 cos x + 6 sinx.
2
2.
11
11463. Find the values of x and y for which cosec 0=
satisfied.
X2 is
11
1147Illustration 4.47 Find the smallest positive values of x and
y satisfying x – y = – and cotx + coty= 2.
11
1148If ( boldsymbol{x}=tan boldsymbol{theta}+cot boldsymbol{theta}, boldsymbol{y}=cos boldsymbol{theta}-sin theta )
then
A. ( x=y )
B. ( frac{1-y^{2}}{2}=frac{1}{x} )
( ^{mathbf{c}} cdot frac{y^{2}-1}{2}=frac{1}{x} )
D. ( frac{1+y^{2}}{2}=frac{1}{x} )
11
1149Show that
( tan 3 x tan 2 x tan x=tan 3 x- )
( tan 2 x-tan x )
11
1150( frac{sin A}{1+cos A}+frac{sin A}{1-cos A} ) is equal to
( A cdot sin A )
B. ( 2 operatorname{cosec} A )
( c cdot cos A )
D. None of these
11
1151100. If x sin a + y sin 2a + z sin 3a = sin 4a,
x sin b + y sin 2b + z sin 3b = sin 4b,
x sin c + y sin 2c + z sin 3c = sin 4c,
then the roots of the equation –
(z-x
= 0, a, b, c nt, are
T 8
a. sin a, sin b, sinc b. cos a, cos b, cos c
c. sin 2a, sin 2b, sin 2c d. cos 2a, cos 26 cos 2c
11
1152Evaluate ( frac{sin ^{2} 63^{circ}+sin ^{2} 27^{circ}}{cos ^{2} 17^{circ}+cos ^{2} 73^{circ}} )11
1153If ( 15 tan ^{2} theta+4 sec ^{2} theta=23 ) then
( tan ^{2} theta=dots )
A ( cdot frac{27}{15} )
B. 45
( c cdot frac{19}{11} )
D.
11
1154Illustration 4.63
< 370/2.
Solve 2 cos²0 + sin 0 < 2, where it/2 s 0
11
1155Solve :-
[
sin 45^{circ}+cos 45^{circ}
]
11
115621.
The number of integral values of k for which the equation 7
cos x +5 sin x =2k + 1 has a solution is
(2002)
(a) 4 (b) 8 (c) 10 (d) 12
11
1157( operatorname{Given} frac{x-x tan ^{2} 30^{circ}}{1+tan ^{2} 30^{circ}}=sin ^{2} 30^{circ}+ )
( 4 cot ^{2} 45^{circ}-sec ^{2} 45^{circ} . ) Then the value of
( boldsymbol{x} )
( A cdot frac{3}{2} )
в. ( frac{5}{2} )
( c cdot frac{9}{2} )
D. none of these
11
1158If ( boldsymbol{A}+boldsymbol{B}=frac{boldsymbol{pi}}{mathbf{3}} ) and ( cos boldsymbol{A}+cos boldsymbol{B}=mathbf{1} )
then which of the following are true:
This question has multiple correct options
A ( cdot cos (A-B)=frac{1}{3} )
B. ( cos (A-B)=-frac{1}{3} )
C ( cdot|cos A-cos B|=sqrt{2 / 3} )
D. ( |cos A-cos B|=frac{1}{sqrt{3}} )
11
115913. Solve the following system of simultaneous equations for
x and y:
4sin x + 31/cosy = 11
5 x 16sin x – 2 x 31/cosy = 2
11
116060. If 0° <e < 90°, the value of
sin e + cos O is
(1) equal to 1
(2) greater than 1
(3) less than 1
(4) equal to 2
11
1161For ( boldsymbol{x} in(mathbf{0}, boldsymbol{pi}), ) the equation ( sin boldsymbol{x}+ )
( 2 sin 2 x-sin 3 x=3, ) has
A. infinitely many solutions
B. three solutions
c. one solution
D. no solution
11
1162Illustration 3.71 Find the value of cos
+ cos
11
1163A unit radian is approximately equal to
A ( cdot 57^{circ} 17^{prime} 43^{prime} )
,
B . ( 57^{circ} 17^{prime} 45 ” )
c. ( 57^{circ} ) 17′ ( 47 ” )
D. ( 57^{circ} 17^{prime} 49^{prime} )
11
1164Solve the following equation:
( cos x cos 2 x cos 3 x=frac{1}{4} )
11
11654. Solve sinx + siny = sin(x + y) and [xl + byl = 1.
44
11
1166( cot B=2 tan (A-B)=>2 tan B+ )
( cot B ) is equal to
( mathbf{A} cdot tan A )
B. ( cot A )
( c cdot 2 tan A )
D. ( 2 cot A )
11
1167Points in which abscissa and ordinate
have different signs will lie in
A. Ist and IIIrd quadrants
B. Illrd and IVth quadrants
c. Ilnd and Illrd quadrants
D. IInd and IVth quadrants
11
11685
Tet a and ß be any two positive values of x for which
nost. I cos x , and 1 – 3 cos x are in G.P. The minimum
value of a-Bis
d. none of these
11
11694. Number of solution(s) of the equation
sin
x
cos 3x
sin 3x
cos 9x
+ cip- in the interval ( 0. A) is
sin 9x
-= 0 in the interval
cos 27 x
11
1170Find the angle measure of 4 radians.
A ( cdot 114.591^{circ} )
В. ( 141.372^{circ} )
c. ( 229.183^{circ} )
D . ( 282.743^{circ} )
E ( .458 .366^{circ} )
11
1171Evaluate:
( frac{sec x^{o}+tan x^{o}}{sec x^{o}-tan x^{o}} )
11
1172Illustration 4.5 Find the values of which satisfy r sin 0=3
and r=4 (1 + sin 8), 0 <O<2n.
11
1173If ( sin theta=cos left(2 theta-45^{circ}right), quad 0<(2 theta- )
( left.45^{circ}right)<90^{circ}, ) then ( tan theta )
A . -1
B.
c.
D. ( frac{1}{sqrt{3}} )
11
1174( frac{tan theta+sec theta-1}{tan theta-sec theta+1}= )
( mathbf{A} cdot frac{cos theta}{1+sin theta} )
B. ( frac{1+cos theta}{sin theta} )
( mathbf{C} cdot frac{1+sin theta}{cos theta} )
D. ( frac{1-sin theta}{cos theta} )
11
11757. Let f(0) = sin 0 (sin 0+ sin 38). Then f (O) is
a. 20 only when 020 b. S 0 for all real 0
c. 20 for all real 0 d. <0 only when oso
11
1176Find the value of
( left(tan 2^{circ} tan 4^{circ} tan 6^{circ}——tan 8right. )
11
1177The angle of the sun above the horizon
is 27.5 degrees. Find the approximate length of the shadow of a person who is
4.75 feet tall.
A . 4.75
в. 2.47
c. 4.65
D. 9.12
E . 4.86
11
1178Prove that ( 2 sec ^{2} theta-sec ^{4} theta- )
( 2 cos e c^{2} theta+cos e c^{4} theta=cot ^{4} theta-tan ^{4} theta )
11
1179If ( 6 sin ^{2} theta-sin theta=1 ) and ( 0 leq theta leq pi )
calculate the value of ( sin theta )
A ( cdot frac{1}{6} )
B.
( c cdot frac{1}{2} )
D. 19
E . 30
11
118064. The general solution of the equation sin100x – cos100x = 1 is
a. 2nd+ , nel b. nt+ – ,ne I
c. nn +
-,nel
ne
d. 2nd – – ,ne I
11
1181Find the solution of
(i) ( 10 cos ^{3} x-16 cos x=3 cos 2 x+3, ) in
the interval ( [-boldsymbol{pi}, boldsymbol{pi}] )
( (i i) 2(cos x+cos 2 x)+sin 2 x(1+ )
( 2 cos x)=2 sin x ) in the interval ( [-pi, pi] )
11
1182General value of ( theta ) satisfying the
eqation ( tan ^{2} theta+sec 2 theta=1 ) is
( mathbf{A} cdot m pi, n pi pm frac{pi}{3}, m, n in I )
В ( cdot m pi, n pi pm frac{pi}{4}, m, n in I )
c. ( m pi, n pi pm frac{pi}{6}, m, n in I )
D. ( m pi, n pi pm frac{pi}{8}, m, n in I )
11
1183Illustration 3.31 Prove that
sin 5A – sin 3A
= tan A
cos 5A + cos 3A
sin A + sin 3A
-= tan 2A
cos A + cos 3A
11
118482. The value of the following is:
(tan 20°)2 (cot 20°)2
(cosec 70°)2 + (sec 70°)2
2tan 15°. tan 45º tan 75°
(1) 1
(2) 4
(3) 3
(4) 2
11
11856. Number of integral value(s) of m for which the equation
4m – 6
sin x – 13 cos x = has solutions, x = [0, 21), is
4 – m
11
1186Solve :
[
tan ^{2} theta-2 sin theta=0
]
11
1187If ( tan B=frac{2 sin A sin C}{sin (A+C)} ) then
( tan A, tan B, tan C ) are in
A . A.
в. G.
c. н.
D. AGP
11
1188Illustration 4.61 If m and n (n > m) are positive integers,
then find the number of solutions of the equation n|sin x1 =
m|cos x| for x E[0, 21). Also find the solution.
11
1189( sec theta-tan theta=6 )
Then ( sec theta+tan theta=? )
11
1190Verify that:
( cos 60^{circ}=frac{1-tan ^{2} 30^{circ}}{1+tan ^{2} 30^{circ}}=frac{1}{2} )
11
1191The radius of circle is ( 9 mathrm{cm} ). Find the
length of an arc of this circle which cuts
off a chord of length equal to the radius.
11
1192Find the value of
( sin 135^{circ} )
11
1193Illustration 4.35
Solve tan 0 + tan 20 + V3 tan 0 tan 20 =
V3
11
1194Prove ( : frac{cos ^{2} theta}{1-tan theta}+frac{sin ^{3} theta}{sin theta-cos theta}=1+ )
( sin theta cos theta )
11
11958. The expression (tan^x + 2 tan²x + 1) cos²x, when x=īt/12,
can be equal to
a. 4(2 – 13)
c. 16 cos? tt/12 d. 16 sin®īt/
12 0
can be equal
b, 46 13
b. 4( N2 +1)
11
11961. If 4 sin4x + cos*x = 1, then x is equal to (n e 2)
a. nt
b. nnt
sin-1
2na
d. 2nd =
11
119711. Eliminate x from equations sin(a + x) = 2b and sin(a – x)
= 2c.
11
1198If ( sec theta=frac{25}{7}, ) then find the value of
( tan theta ? )
11
11999. Solve tan ( coso – cot ( sin o)11
1200In ( triangle A B C, ) if ( a, b, c ) are in ( A . P . ) then ( cot frac{boldsymbol{A}}{2} cot frac{boldsymbol{C}}{2}= )
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot 3 )
D.
11
120110. The value of 0 € (0,21) for which 2 sine – 5 sin 0+2>
O is
417
(IIT-JEE 2006
11
1202If ( 0<x leq frac{pi}{2}, ) then ( sin x+operatorname{cosec} x geq )
( mathbf{A} cdot mathbf{0} )
B.
( c cdot 2 )
D. 3
11
1203Let ( X ) be the solution set of the equation ( boldsymbol{A}^{x}=boldsymbol{I}, ) where ( boldsymbol{A}=left[begin{array}{ccc}mathbf{0} & mathbf{1} & mathbf{- 1} \ mathbf{4} & mathbf{- 3} & mathbf{4} \ mathbf{3} & mathbf{- 3} & mathbf{4}end{array}right] ) and
is the corresponding unit matrix and
( boldsymbol{x} subseteq boldsymbol{N}, ) then the minimum value of
( sumleft(cos ^{x} theta+sin ^{x} thetaright), theta in R )
11
1204f ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=boldsymbol{x} boldsymbol{y} boldsymbol{z}, ) prove that
( frac{2 x}{1-x^{2}}+frac{2 y}{1-y^{2}}+frac{2 z}{1-z^{2}}= )
( frac{2 x}{1-x^{2}} frac{2 y}{1-y^{2}} frac{2 z}{1-z^{2}} )
11
1205If ( boldsymbol{A}-boldsymbol{B}=boldsymbol{C} ) and ( boldsymbol{A}+boldsymbol{B}=frac{pi}{2} ) then
( tan A=tan B+2 tan C )
11
1206+1+1+1 is
+-
6. The value of – -+ –
tan a tan ß
tand
a. – 8
c. 2/3
tan y
b. 8
d. 1/3
11
120792. If cos x + cos y – cos(x + y) =
=
=
2
th
a. x + y = 0
c. x=y
b. x = 2y
d. 2x = y
11
1208Find the general solution of the equation ( tan ^{2} alpha+2 sqrt{3} tan alpha=1 )11
120928. Iff(x) = cos_0+ sec 6, then
a. f(x) f(x) > 1
d. f(x) 22
11
1210Tllustration 3.64 Prove that tan — is a root of polynomial
equation 5×4 – 10x² + 1 = 0.
10
11
1211Solve the following equation:
( tan x=-1 )
11
1212Express the following angle in degree ( left(-frac{7 pi}{12}right)^{c} )11
121345. If a, b, y, are the smallest positive angles in ascending
order of magnitude which have their sines equal to the
positive quantity k, then the value of 4 sin + 3 sin Þ
+2 sin
+ sin
is equal to
2
b. 2/1+k
a. 2/1-k
a Vith
d. none of these
11
1214If ( boldsymbol{alpha} ) and ( beta ) are two different values of ( boldsymbol{theta} )
lying between 0 and ( 2 pi ) which satisfy
( 3 cos theta+4 sin theta=6 . ) Find the value of
( sin (alpha+beta) )
11
121516. Let f (x) = ab sin x + b V1-a² cos x+c, where |al 0 then
a. maximum value of f(x) is b if c = 0
b. difference of maximum and minimum values of f(x)
is 2b
g
o
c. f(x) = c if x = – cos-1
a
d. f(x) = c if x = cos’ a
11
1216Illustration 3.48 If p(x) = sin x (sinºx+3)+cosx (cos’x +4)
+ (1/2) sin 2x + 5, then find the range of p(x).
11
1217The area of the circle is ( 25 pi ) sq. cms.
Find the length of its arc subtending an angle of ( 144^{circ} ) at the centre. Also find the
area of the corresponding sector.
11
1218Find the number of solutions of the
equations;
( |cot x|=cot x+frac{1}{sin x}, ) where ( x in )
( [0,2 pi] )
11
1219( fleft(m^{2} cos frac{2 pi}{15} cos frac{4 pi}{15} cos frac{8 pi}{15} cos frac{14 pi}{15}=n^{2}right. )
then find the value of ( frac{m^{2}-n^{2}}{n^{2}} )
11
1220Evaluate :
( frac{2 cos 67^{circ}}{sin 23^{circ}}-frac{tan 20^{circ}}{cot 50^{circ}}-cos 0^{circ} )
11
1221Solve:
( frac{x sin x}{1+cos x} )
11
1222Solve:
( 3 sin ^{2} x-7 sin x+2=0 )
11
1223A flag-staff stands on a tower which is on level ground. The total height of the flag-staff and tower taken together is
300 metres. The flag-staff subtends an angle of ( tan ^{-1}left(frac{1}{5}right) ) at a point ( P ) on the level ground at a distance 300 metres from the foot of the tower. The height of the tower is:
A. 100 metres
B. 200 metres
c. 250 metres
D. 300 metres
11
1224The value of ( sin 51^{circ}+cos 81^{circ} ) is
A . ( cos 21^{circ} )
B. ( sin 21^{circ} )
( mathbf{c} cdot cos 42^{circ} )
( mathbf{D} cdot sin 42^{circ} )
11
12259.
Let A and B denote the statements
A: cos a + cos B + cos y=0
B: sin a + sin ß+sin y=0
If cos (B-Y)+cos (y-a)+cos (a-B)= , then : [2009
(a) A is false and B is true (b) both A and B are true
(c) both A and B are false (d) A is true and B is false
11
1226Illustration 3.80 If A + B + C = Tt, prove that cot A + cot B
+ cot C – cosec A. cosec B. cosec C = cot A . cot B . cot C.
11
122714. The number of solutions of equation 6 cos 20+2 cos?(@/2)
+ 2 sin?O=0,-< < it is
a. 3
b. 4
c. 5
d. 6
11
1228Illustration 3.74 Prove that 2 sin 2° + 4 sin 4° + 6 sin 6° +
… + 180 sin 180º = 90 cot 10°.
Ilustration 3: 75 prove that 2 sin 2° + sin 4 + 6 sin 6° +
11
122910. The value of (tan? “* + tan ? 2 + tan? 34) (cov? ”
+ cot2 217 + cot? 37 is
a. 105
c. 210
b. 35
d. none of these
11
1230Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, shall be three times as old as you will be.” Is not this interesting? Represent this situation algebraically.11
1231( sin 1^{circ} sin 2^{circ} sin 3^{circ} dots dots sin 180^{circ} ) is equal
to
( mathbf{A} cdot mathbf{1} )
B. –
c. 0
D. ( frac{1}{2} )
11
123214. If sinx – a sin x + b = 0 has only one solution in (0,7),
then which of the following statements are correct?
a. a E (-∞,1] [2,-) b.be (-0,0] [1, )
c. a= 1 + b
d. none of these
11
1233( mathbf{0}<mathbf{x}<mathbf{2} boldsymbol{pi} ; mathbf{0}<mathbf{y}<mathbf{2} boldsymbol{pi} ) and
( 3^{sin x+cos y}=1 ) and ( 25^{sin x^{2}+cos ^{2} y}=5 ) then
( (x, y) ) is This question has multiple correct options
( ^{mathrm{A}} cdotleft(frac{7 pi}{6}, frac{pi}{3}right. )
В. ( left(frac{7 pi}{6}, frac{5 pi}{3}right) )
c. ( left(frac{11 pi}{6}, frac{pi}{3}right) )
D. ( left(frac{11 pi}{6}, frac{5 pi}{3}right) )
11
1234Illustration 6.21 Solve V5 –14 V10+245 = 8,76(0,5).
sin x
COS X
11
123565. If seco + tano = 5, then the value
tan 0+1
is
tan 0 – 1
11
1236If ( tan theta=frac{1}{2} ) and ( tan phi=frac{1}{3}, ) then the
value of ( boldsymbol{theta}+boldsymbol{phi} ) is
A ( cdot frac{pi}{6} )
в. ( pi )
c. ( frac{pi}{4} )
D.
11
1237ULL
17.
Let f(O)=sin(sino+sin30). Then f (0) is (2000)
(a) 20 only when 020 (6) < 0 for all real 0
© 20 for all real e (d) <0 only when 0 <0
11
1238The sum of all the solutions of the
equation ( cos theta cos left(frac{pi}{3}+thetaright) cos left(frac{pi}{3}-thetaright)= )
( frac{1}{4}, theta epsilon[0,6 pi] )
A. ( 15 pi )
в. ( 30 pi )
c. ( frac{100 pi}{3} )
D. None of these
11
1239Fin an acute angle ( Theta ), when
( frac{cos Theta-sin Theta}{cos Theta+sin Theta}=frac{1-sqrt{3}}{1+sqrt{3}} )
11
124013. Which of the following is not the general solution of
2cos2x + 1 = 3.2-sin?x?
a. nn, ne Z
b. n + – 1 ,ne Z
2)
it , ne z
d. none of these
2
11
12414. In A ABC, if sin’e = sin(A – ) sin(B – 0) sin(C – 0), then
prove that cot 0= cot A + cot B + cot C.
11
1242From the following exact of the sine
table, the value of ( sin 37^{0} 27^{prime} ) is equal to
( 12^{prime} 18^{prime} 24^{prime} 30^{prime} 36^{prime} 42^{prime} 48^{prime} 54^{prime} 1^{prime} 2 )
A . 0.6075
B. 0.6081
c. 0.6088
D. 0.6115
11
124318. If 3 sin ß= sin (2a + B), then tan (a +B) – 2 tan a is
Tot a. independent of a
b. independent of ß
c. dependent of both a and
B a r
d. independent of both a and ß
11
1244If ( cos A=frac{4}{5} ) find ( tan A )11
1245The solution of the equation
( (sin x+cos x)^{1+sin 2 x}=2,-pi leq x leq pi )
is
A ( cdot frac{pi}{2} )
B.
c. ( frac{pi}{4} )
D. none of these
11
124647. If tan 30+ tan 0 = 2 tan 20, then 0 is equal to (ne zo
na
no a. nn OOTDT b.
d. none of these
11
124755.
If tan(a coso) = cot (nt sino), then
will be equal to (O se
11
1248Find the degree measures corresponding to the following radian measures ( left(boldsymbol{U s e} boldsymbol{pi}=frac{mathbf{2 2}}{mathbf{7}}right) )
(i) ( frac{11}{16} )
(ii) -4
(iii) ( frac{5 pi}{3} )
(iv) ( frac{7 pi}{6} )
11
124943. If cos 25° + sin 25º = p, then cos 50° is
a. 12 – p? b. – √2-p²
c. p /2-p2. d.-pſ2-p²
rata
11
1250If ( 0 leq x leq pi, ) the interval in which the
function ( f(x)=frac{tan x}{sin x} ) is defined is:
A. ( 0 leq x leq pi )
В. ( 0<x<pi )
c. ( 0<x<frac{pi}{2} )
D ( cdot frac{pi}{2}<x leq pi )
E ( cdot 0<x<frac{pi}{2} ) and ( frac{pi}{2}<x<pi )
11
1251Find the value of ( frac{cos A-sin A+1}{cos A+sin A-1}- )
( (operatorname{cosec} A+cot A) )
11
1252Assertion
The number of real solution of the
equation ( sin (cos x)=cos (sin x) ) is
zero
Reason
( sin x>0, ) then ( 2 n pi<x< )
( (2 n+1) pi, n epsilon I )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion false but Reason is true
11
125311. Let tanx-tan-x >0 and 2sin x nt,ne
Z b . x >nt – Tc/6, n e Z
c. x<nt – Te/4, ne Z d. x<nt + tc/6, ne z
11
125448. If cose sin € + V sin’e + sin’a} Sk, then the value of k
is
on
a. Vi+cos’
a
b
d
. V1+sinʼa
. √2 + cos²a
c. √2+ sin²a
11
1255Evaluate: ( int_{0}^{frac{pi}{2}} sqrt{1+sin x} d x )11
1256x +

7. The number of solutions of the equatio
CO.
+ cos x – 2 cos (x+”). cos* = sin? – in interval
11
1257Find the value of ( cos theta-cos 3 theta+cos 4 theta )
when ( boldsymbol{theta}=mathbf{4 5}^{circ} )
A. ( frac{sqrt{3}}{2} )
B. ( sqrt{3}-1 )
( c cdot frac{1}{2} )
D. ( sqrt{2}-1 )
11
1258The possible value of ( boldsymbol{theta} in(mathbf{0}, boldsymbol{pi}) ) such
( operatorname{that} sin (theta)+sin (4 theta)+sin (7 theta)=0 ) are
A ( cdot frac{2 pi}{9}, frac{pi}{4}, frac{4 pi}{9}, frac{pi}{2}, frac{3 pi}{4}, frac{8 pi}{9} )
В. ( frac{pi}{4}, frac{5 pi}{12}, frac{pi}{2}, frac{2 pi}{3}, frac{3 pi}{4}, frac{8 pi}{9} )
С ( cdot frac{2 pi}{9}, frac{pi}{4}, frac{pi}{2}, frac{2 pi}{3}, frac{3 pi}{4}, frac{35 pi}{36} )
D. ( frac{2 pi}{9}, frac{pi}{4}, frac{pi}{2}, frac{2 pi}{3}, frac{3 pi}{4}, frac{8 pi}{9} )
11
1259( tan theta=-2, theta epsilon(0, pi) ) then which of the
following is correct
This question has multiple correct options
A ( cdot sin theta=frac{-2}{sqrt{5}} )
B. ( sin theta=frac{2}{sqrt{5}} )
( ^{mathrm{C}} cdot cos theta=frac{-1}{sqrt{5}} )
D. ( cos theta=frac{1}{sqrt{5}} )
11
1260Find the value of ( cos 37^{circ} 16^{prime} )11
1261Find the centroid of Triangle whose
vertices are
( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}(mathbf{2},-mathbf{1}, mathbf{6}), boldsymbol{C}(mathbf{3}, mathbf{2},-mathbf{3}) )
A ( .(2,1,2) )
В. (2,-1,2)
c. (-2,-2,-2)
D. (1,2,1)
11
126290. If cos A + cos²B + cos²C = 1, then A ABC is
a. equilateral
b. isosceles
c. right angled
d. none of these
11
1263Which of the following statements is/are correct for ( 0<theta<frac{pi}{2} ? )
This question has multiple correct options
A ( cdot(cos theta)^{1 / 2} leq cos frac{theta}{2} )
B. ( (cos theta)^{3 / 4} geq cos frac{3 theta}{4} )
c. ( cos frac{5 theta}{6} geq(cos theta)^{5 / 6} )
D. ( cos frac{7 theta}{8} geq(cos theta)^{7 / 8} )
11
1264Solve ( sin (50+theta)-cos (40-theta)+ )
( tan 1 tan 10 )
( tan 20 tan 70 tan 80 tan 89=1 )
11
1265If ( tan 6 theta=frac{p}{q}, ) find the value of ( frac{1}{2}(p operatorname{cosec} 2 theta-q sec 2 theta) ) in terms of ( p )
and ( boldsymbol{q} )
( mathbf{A} cdot 2 sqrt{p^{2}+q^{2}} )
B ( cdot sqrt{p^{2}+q^{2}} )
c. ( frac{sqrt{p^{2}+q^{2}}}{q} )
D. ( frac{sqrt{p^{2}+q^{2}}}{p} )
11
1266If (x – a) cos 0 + y sin 0= (x – a) cos 0 + y sin o = a and
tan (8/2) – tan (0/2) = 2b, then
a. y2 = 2ax – (1 – 62) x2
b. tan = 6 + bx)
² = 26x- (1 – 2) x²
d. tan 2-1 (y – bx)
2 x
2.
X
11
1267The value of ( int_{0}^{frac{pi}{2}} frac{cos 3 x+1}{2 cos x-1} d x ) is
( A cdot 2 )
B.
( c cdot frac{1}{2} )
D.
11
12683. Prove that 5 cos 0 +
8 +-
+ 3 lies between – 4
(IIT-JEE 1979)
and 10.
11
12692. The maximum value of the expression
-is (IIT-JEE 2010)
sin? 0 + 3 sin o cos 0 + 5 cos? O
11
1270If
( boldsymbol{theta} ) and ( phi ) are angles in the 1 st quadrant such that ( tan theta=frac{1}{7} ) and ( sin phi=frac{1}{sqrt{10}} )
( mathbf{A} cdot theta+2 phi=90^{circ} )
B . ( theta+2 phi=60^{circ} )
c. ( theta+2 phi=30^{circ} )
D. ( theta+2 phi=45^{circ} )
11
1271Prove that:
( tan x+tan left(frac{pi}{3}+xright)-tan left(frac{pi}{3}-xright)= )
( 3 tan 3 x )
11
1272Find the value of ( tan ^{-1}left(tan frac{2 pi}{3}right) )11
1273Find ( ^{prime} x^{prime} ) if ( sec ^{2} 2 x=1-tan 2 x )11
1274Find :
( cos frac{pi}{65} cos frac{2 pi}{65} cos frac{4 pi}{65} cos frac{8 pi}{65} cos frac{16 pi}{65} )
A ( cdot frac{1}{64} )
в. ( frac{1}{32} )
( c cdot frac{1}{16} )
D. None of these
11
127520. In a AABC, if tan A : tan B : tan C = 3: 4: 5, then the
value of sin A sin B. sin C is equal to
b. 215
T5
c. 215
d.
375
11
1276n a right angled ( triangle A B D, angle B= )
( mathbf{6 0}^{circ}, angle boldsymbol{A}=mathbf{3 0}^{circ} )
Then ( sin 30^{circ} ) is equal to
( A )
( 3 frac{1}{2} )
( c )
( D cdot 1 )
11
127717. The number of solutions of the equation 1 + cos x +
cos 2x + sin x + sin 2x + sin 3x = 0, which satisfy the
condition
sa is
11
1278Illustration 3.29
Prove that cos 55° + cos 65° + cos 175° = 0.
11
127962. If OSO 90° and cos20-sin30
= cos 90°, then will be equal
to
(1) 16° (2) 18°
(3) 20°
(4) 22°
11
1280The value of ( frac{sin ^{2} 53+cos ^{2} 53}{sec ^{2} 37-tan ^{2} 37} ) is
( mathbf{A} cdot mathbf{1} )
B. 2
( c cdot frac{1}{4} )
D.
11
1281Prove that: ( sec ^{6} boldsymbol{A}-tan ^{6} boldsymbol{A}=mathbf{1}+ )
( mathbf{3} tan ^{2} boldsymbol{A}+mathbf{3} tan ^{mathbf{4}} boldsymbol{A} )
11
1282( 160^{circ} ) in radian measure is
A ( cdot frac{2 pi^{c}}{5} )
в. ( frac{3 pi^{c}}{5} )
c. ( frac{8 pi^{c}}{9} )
D. ( pi c )
11
1283If ( sin (theta+alpha)=a ) and ( sin (theta+beta)= )
( boldsymbol{b},((mathbf{0}<boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{theta}<boldsymbol{pi} / 2)) ) then
( 2 cos ^{2}(alpha-beta)-1-4 a b cos (alpha-beta) ) is
A ( cdot 1-a^{2}-b^{2} )
B . ( 1-2 a^{2}-2 b^{2} )
c. ( 2+a^{2}+b^{2} )
D . ( 2=a^{2}-b^{2} )
11
1284ootan
99. The value of a
norna
o
cos
in-
2
1
1
b.
b._2
sin 20
ē
. sin 200
C. sin 20
0
d. sin o
ő
11
1285Illustration 3.57
If tan
olm
| 01
tan
ola
prove that
Vatb
cosa=
a cos o +b
a+bcos o
11
1286Consider the geometric progression ( boldsymbol{S}=mathbf{1}+boldsymbol{2} sin ^{2} boldsymbol{theta}+boldsymbol{4} sin ^{4} boldsymbol{theta}+boldsymbol{8} sin ^{6} boldsymbol{theta}+ )
( ldots . ) up to infinite terms, where ( mathrm{S} ) is a finite number and ( boldsymbol{theta} neq frac{boldsymbol{n} boldsymbol{pi}}{mathbf{2}} ) where ( mathbf{n} varepsilon mathbf{I} )
Then Values of ( theta ) always lies in the
interval?
A ( cdotleft(-frac{pi}{6}, frac{pi}{6}right) )
в. ( left(0, frac{pi}{3}right) )
c. ( left(-frac{pi}{3}, 0right) )
D. ( left(-frac{pi}{4}, frac{pi}{4}right)-{0} )
11
1287Value of the expression ( (1-cos Theta)(1+ )
( cos Theta)left(1+cot ^{2} Thetaright) ) is
A .
B.
( c cdot sin ^{2} theta )
D. ( operatorname{cosec}^{2} Theta )
11
128870. The value of sin2 65° + sin2 25°
+ cos2 35° + cos2 55° is
(1) O
(2) 1
(3) 2
11
1289Illustration 4.52
Solve cos5°x – sin50x = 1.
11
1290If ( a=frac{sin x times cos 3 x}{sin 3 x times cos x} ) then which of the
following is wrong?
A ( cdot a3 )
c. ( frac{1}{3}<a<3 )
D. None of these
11
1291Express the ( 150^{0} ) in radians:11
12927. If tan 60=plq, find the value of -(p cosec 20 – q sec 20)
in terms of p and q.
11
1293Find the value of ( sin ^{2} 30^{circ}+cos ^{2} 60^{circ} )11
1294The simplification of ( cos (A+ )
( B) cos (A-B) ) is equivalent to:
A ( cdot sin ^{2} A-sin ^{2} B )
B. ( cos ^{2} B-sin ^{2} A )
( mathbf{c} cdot cos ^{2} A-cos ^{2} B )
D. ( cos 2 A cdot sin 2 B )
11
1295If ( tan alpha=2, ) then the value of
( frac{sin alpha}{sin ^{3} alpha+cos ^{3} alpha} ) is
( A cdot frac{2}{9} )
в. ( frac{5}{9} )
c. ( frac{10}{9} )
D. ( frac{5.5}{9} )
11
129610. If x + y = 27/3 and sin x/sin y = 2, then the
a. number of values of x € [0, 41] are 4
b. number of values of x € [0, 41] are 2
c. number of values of ye [0, 41] are 4
d. number of values of ye [0, 41] are 8
11
1297n the given figure, ( angle boldsymbol{A}+angle boldsymbol{B}+angle boldsymbol{C}+ )
( angle D+angle E ) is equal to
2
( c cdot frac{3 pi}{pi} )
2
22
11
129828. The value of sin? 12° + sin? 21° + sin? 39° + sin? 48° –
sin? 9º – sin? 18° is
11
1299Illustration 3.16
Prove that tan 70° = 2tan 50° + tan 20°.
11
1300Illustration 2.16 By geometrical interpretation, prove that
i. sin(a+B) = sin a cos ß+ sin ß cos a
ii. cos(a + B) = cos a cos B- sin a sinß
11
130152. If 2 sin 20|=|tan B + cot Bl, a,ße 69,
value of a + Bis
then the
b. T
niel
d.
11
1302Find the value of ( frac{cos 70^{0}}{sin 20^{0}}+ )
( cos 57^{0} operatorname{cosec} 33^{0}-2 cos 60^{0} )
11
1303If ( alpha, beta, gamma, delta ) are in arithmetic
progression. Then which is of the following is correct
A. ( tan (alpha+delta)=tan (beta+gamma) )
B. ( tan (alpha+gamma)=tan (beta+delta) )
c. ( tan (alpha+beta)=tan (gamma+delta) )
D. none of these
11
13042. Number of roots of the equation sinx cos xl +
| 2 + tanx + cotx = 3 ,xe 0, 41, are
11
13054xy
15. sec 0 =
(x + y)2 15
2 is true if and only if (1996 – 1 Mark
(a) x+y=0
(b) x=y,x=0
(d) x 70, y0
(c) x=y
11
1306The value of the expression ( left(1+cos frac{pi}{10}right)left(1+cos frac{3 pi}{10}right)left(1+cos frac{7 pi}{10}right)(1 )
is
11
1307( mathbf{A}=cos 20^{0} cos 40^{0} cos 60^{0} cos 80^{0} )
( mathbf{B}=cos 6^{0} cos 42^{0} cos 66^{0} cos 78^{0} )
( mathbf{C}=cos mathbf{3} mathbf{6}^{mathbf{0}} cos mathbf{7} mathbf{2}^{mathbf{0}} cos mathbf{1 0} mathbf{8}^{mathbf{0}} cos mathbf{1} mathbf{4} mathbf{4}^{mathbf{0}} )
A. ( A>B>C )
в. ( B>C>A )
c. ( C>A>B )
D. ( A=B=C )
11
1308Solve the following equations. ( sin frac{x}{2} cos 2 x+sin ^{2} x cos frac{x}{2}= )
( cos ^{2} x cos frac{x}{2} )
11
1309If ( theta ) is in the first quadrant and ( cos theta= )
( frac{3}{5}, ) then the value of ( frac{5 tan theta-4 operatorname{cosec} theta}{5 sec theta-4 cot theta} ) is
A ( cdot frac{5}{34} )
в. ( frac{5}{16} )
c. ( frac{5}{-34} )
D. ( frac{-5}{16} )
11
1310If the tangents of the angles ( A ) and ( B ) of ( a ) triangle ( A B C ) satisfy the equation
( a b x^{2}-c^{2} x+a b=0, ) then
This question has multiple correct options
( mathbf{A} cdot tan A=frac{a}{b} )
B. ( tan B=frac{b}{a} )
( mathbf{c} cdot cos C=0 )
D. ( sin ^{2} A+sin ^{2} B+sin ^{2} C=2 )
11
1311Solve the following equations. ( cos ^{6} x-sin ^{6} x=frac{13}{8} cos ^{2} 2 x )11
1312Find the value of ( tan 25^{circ} 15^{prime} )11
1313Find the distance between ( mathrm{P}( )
( a sin alpha,-b cos alpha) & Q(-a cos alpha, b sin alpha) )
11
1314The number of solution of tan x+sec x=2cos x in [0,2 T) is
[2002]
(a) 2 (b) 3 C) 0 (d) 1
11
1315Find the values of other trigonometric
function ( (sin , cos , tan ) ) in each of the
following problems
i) ( sin theta=3 / 5: theta ) in ( I^{s t} ) quadrant
ii) ( cos theta=-1 / 2: theta ) in ( I I^{s t} ) quadrant
iii) ( sin theta=3 / 4: theta ) in ( I I I^{s t} ) quadrant ( $ $ )
11
1316( A ) tower of ( x ) meters high has a flagstaff at its top. The tower and the flagstaff subtend equal angles at a point distant
( y ) meters from the foot of the tower then
the length of the flagstaff in meters is
A ( frac{yleft(x^{2}-y^{2}right)}{x^{2}+y^{2}} )
B. ( frac{xleft(y^{2}+x^{2}right)}{y^{2}-x^{2}} )
c. ( frac{xleft(x^{2}+y^{2}right)}{x^{2}-y^{2}} )
D. ( frac{xleft(x^{2}-y^{2}right)}{x^{2}+y^{2}} )
11
1317If ( P=cos frac{pi}{20} cdot cos frac{3 pi}{20} cdot cos frac{7 pi}{20} cdot cos frac{9 pi}{20} & Q )
( =cos )
( frac{pi}{11} cdot cos frac{2 pi}{11} cdot cos frac{4 pi}{11} cdot cos frac{8 pi}{11} cdot cos frac{16 pi}{11}, ) ther
is
A. not defined
B. 1
( c cdot 2 )
D. none of these
11
1318Illustration 3.75 If A + B + C = 180°, prove that cos-A +
cos-B + cos²C= 1-2 cos A cos B cos C.
COS
11
1319If ( 0, alpha, beta<frac{pi}{4} ) such that ( cos (alpha+beta)=frac{4}{5} )
andsin ( (alpha-beta)=frac{5}{13}, ) then the value of
( tan 2 alpha= )
A. ( frac{56}{65} )
в. ( frac{56}{23} )
c. ( frac{56}{33} )
D. ( frac{56}{36} )
11
13201-
X
then value of
57. If tan 62° = 1+ x
tan 208° will be
(1)
1-x²
1+ 2
1 + x2
1-x²
1
X
+
x
11
1321Illustration 4.38
Find the number of roots of the equation
tan x+3) = 2 tan x, for IE (0, 3m).
11
1322( frac{sin 30^{0}+tan 45^{0}+operatorname{cosec} 60^{0}}{sec 30^{0}+cos 60^{0}+cot 45^{0}}= )11
1323Find the value of ( 2 sin 3 theta cos theta- )
( sin 4 theta-sin 2 theta )
11
1324IF ( theta ) is in the first quadrant and ( cos theta= )
3. then value of
( overline{mathbf{5}} )
( 5 tan theta-4 operatorname{cosec} theta sec theta-4 cot theta ) is
A ( cdot frac{-14}{3} )
в. ( frac{5}{16} )
( c cdot frac{5}{-34} )
D. ( frac{-5}{16} )
11
1325=1, then AABC is
+tan 2 TI-B
– + tan 2 T-C
13. If tan21-A
4 ou 4
a. equilateral
ollers c. scalene
b. isosceles
d. none of these 2011 SS
11
1326( ln operatorname{acircle} ) of radius ( 21 mathrm{cm}, ) an arc
subtends an angle of ( 60^{circ} ) at the
centre.Find
(i) the length of the arc
( (i i) ) the area of the sector
(iiii) the area of the minor segment and
( (i v) ) the area of the major segment.
11
1327Illustration 3.74 Prove that 2 sin 2° + 4 sin 4° + 6 sin 6° +
… + 180 sin 180° = 90 cot 10°.
11
1328( (sin theta+csc theta)^{2}+(cos theta+sec theta)^{2} ) is
( A cdot geq 9 )
B . ( leq 9 )
( mathrm{c} cdot=9 )
D. None of these
11
1329If ( boldsymbol{A}=mathbf{6 0}^{circ} ) and ( boldsymbol{B}=mathbf{3 0}^{boldsymbol{o}}, ) verify that
( tan (boldsymbol{A}+boldsymbol{B})=frac{tan boldsymbol{A}+tan boldsymbol{B}}{1-tan boldsymbol{A} tan boldsymbol{B}} )
11
133029. Value of 3+cot 80% cot 20°
cot 80°+cot 20°
o is equal to
a. cot 20°
b. tan 50°
c. cot 50°
d. cot 20°
11
1331The value of ( x ) fo which ( sin (pi x)+ )
( cos (pi x)=0 )
11
1332If ( cos theta+sin theta=sqrt{2}, ) find the value of
( cos theta-sin theta )
11
133348. The solution of 4 sin’x + tan-x + cosec?x + cotx – 6=0
is (n e 2)
a. nt +
b. 2nd
Bit Bim
c. Na +
d. nr –
11
1334( I f cos 10^{circ} cos 30^{circ} cos 50^{circ} cos 70^{circ}=x )
Find ( 16 x )
11
1335The number of solution of the equation ( |cot x|=cot x+frac{1}{sin x} operatorname{in}[0,2 pi] ) is
( A cdot 2 )
B. 4
c. 0
( D )
11
133621. If tan x + tan 2x + tan 3x = tan x tan 2x tan 3x then value
of sin 3x + cos 3x| is
11
1337If ( sin theta=0.47, ) then ( sin (pi-theta)= )
A . -0.47
B . -0.43
c.
D. 0.43
E . 0.47
11
1338Illustration 3.47 Let f(x) = 2 cosec 2x + sec x + cosec x. Then
find the minimum value of f(x) for x el
11
133933. General solution of sinx – 5 sinx cos x – 6 cos²x = 0 is
a. x=nt – Te/4, n e Z only
b. nt + tan-‘ 6, ne Z only
c. both (a) and (b)
o consavia
d. none of these
11
1340Solve the equation ( sin theta+sin 3 theta+sin 5 theta=0 )11
13414. If cot(a + B) = 0, then sin(a + 2B) can be
a. – sin a
b. sin ß
c. cos a
d. cos ß
11
13423. Sum of roots of the equation x4 – 2x² sin? * + 1 = 0 is
a. O
c. 1
b. 2
d. 3
11
13432. For which values of a does the equation 4 sin(x + Tt/3)
cos(x – /6) = 0? + V3 sin 2x – cos 2x have solutions?
Find the solutions for a = 0, if any exists.
11
1344The measure of an angle in degrees,
grades and radians be ( mathrm{D}, mathrm{G} ) and ( mathrm{C} )
respectively, then relation between them ( frac{boldsymbol{D}}{mathbf{9 0}}=frac{boldsymbol{G}}{mathbf{1 0 0}}=frac{boldsymbol{2 C}}{boldsymbol{pi}} ) but ( mathbf{1}^{circ}= )
( left(frac{180}{pi}right)^{0} simeq 57^{circ}, 17^{prime}, 44.8^{prime prime} ) and sum of
interior angles of a ( n ) -sided regular polygon is ( (2 n-4) frac{pi}{2} . ) On the basis of above information, answer the following
questions :The angles between the hour hand and minute hand of a clock at half
past three is –
( A cdot frac{pi}{3} )
B. ( frac{pi}{4} )
c. ( frac{5 pi}{12} )
D. ( frac{7 pi}{12} )
11
134517.
Prove that the values of the function
ne
sin x cos 3x
sin 3 x cos x
between and 3 for any real:
100% Made
between
and 3 for any real x.
(1997 – 5 Marks)
11
1346( cos ^{3} theta+cos ^{3}left(120^{circ}+thetaright)+cos ^{3}left(120^{circ}-right. )
( boldsymbol{theta})= )
( A cdot frac{3}{4} sin 3 theta )
B . ( frac{3}{4} cos 3 theta )
( mathrm{c} cdot frac{3}{4} tan 3 theta )
D. ( frac{3}{4} cot 3 theta )
11
134711.
If A=sinx + cos4x, then for all real x:
[2011
(b) 1SAS2
(a) sasi
11
13481. If f(0) =
1-sin 20 + cos 20
-, then value of
2 cos 20
8f (11°) • f (34°) is
11
1349The number of solutions of the equation
( sin x=cos 3 x ) in ( [0, pi] ) is
A . 1
B . 2
( c .3 )
D.
11
135044. The number of roots of (1 – tan 2) (1 + sin 20) = 1 + tan e
for 0 [0, 21] is
a. 3
d. none of these
b. 4
c. 5
11
1351Find the angle made by a ladder of length ( 4 m ) with the ground if its one end is ( 2 m ) away from the wall and the other end is on the wall11
135289.
sin 2 A+sin 2B + sin 2C
is equal to be
sin A+sin B + sin C
A B C abs
a. 8sin
sin – sin b. 8 cos -COS
Ola Ola
18
c. 8 tan
– tan
– tan
8 cot — cot –
11
1353if ( tan 25^{circ}=x, ) then
( frac{tan 155^{circ}-tan 115^{circ}}{1+tan 155^{circ} tan 115^{circ}} ) is equal to
A ( cdot frac{1-x^{2}}{2 x} )
в. ( frac{1+x^{2}}{2 x} )
c. ( frac{1+x^{2}}{1-x^{2}} )
D. ( frac{1-x^{2}}{1+x^{2}} )
11
135431. The number of solutions of the equation sin’x cos x +
sin?x cos²x + sin x cos x = 1 in the interval [0, 271) is/are
b. 2
c. 3
end. infinite
a. 0
11
1355sin a + sin B = and cos a + cos B =
7. The value of sin(a+B) is
d. none of these
11
135613. For x e(0, Tt), the equation sin x + 2 sin 2x – sin 3x = 3
has
ban S
sur
a. infinitely many solutions
b. three solutions
c. one solution
d. no solution
(JEE Advanced 2014)
11
1357What is most general value of ( theta ) which
satisfies both the equations. ( sin theta=-1 / 2 ) and ( tan theta=1 / sqrt{3} )
11
1358General solution of ( sin ^{3} x+cos ^{3} x+ )
( frac{3}{2} sin 2 x=1 )
A. ( x=n pi ) when ( n ) is even integer
B. ( x=2 n pi ) when ( n ) is odd integer
C ( cdot x=n pi+frac{pi}{2} ) when ( n ) is odd integer
D・ ( x=n pi-frac{pi}{2} )
11
135969. If O be acute and tan 0 + cot 0 =
2, then the value of tans 0 + cot10
O is
(1) 1
(2) 2
(3) 3
(4) 4
11
1360Illustration 3.45 If sin A = 3/5 and 0° < A < 90°, find the
values of sin 2A, cos 2A, tan 2A, and sin 4A.
11
136116. If 0 < x < 21, then the number of real values of x, which
satisfy the equation cos x + cos2x+cos 3x + cos 4x =0 is:
[JEEM 2016
(2) 7
(b) 9
(c) 3
(d) 5
1.
fon Av
.
11
1362Illustration 4.31 Solve the equation 2(cosx + cos 2x) + sin
2x(1+2 cosx) = 2 sinx for x (-10 < x < 0).
11
1363If ( 2 sin theta+1=0 ) and ( sqrt{3} tan theta=1 ) then
find general value of ( boldsymbol{theta} ) is
( ^{mathrm{A}} cdot_{n pi pm} frac{pi}{6} )
в. ( quad n pi+(-1)^{n} cdot frac{7 pi}{6} )
c. ( _{2 n pi+frac{7 pi}{6}} )
D. ( 2 n pi+frac{11 pi}{6} )
11
1364The general solution of ( tan x-sin x= )
( 1-tan x sin x )
A ( cdot x=n pi+frac{pi}{4} )
( x=n pi+(-1)^{n}left(-frac{pi}{2}right) )
В ( cdot x=frac{n pi}{4}-frac{pi}{4} )
( x=n pi+(-1)^{n}left(-frac{pi}{2}right) )
C ( cdot x=n pi+frac{pi}{4} )
D. ( x=n pi+frac{pi}{6} )
( x=n pi+(-1)^{n}left(-frac{pi}{2}right) )
11
1365If ( mathbf{p}_{1}, mathbf{p}_{2}, mathbf{p}_{3} ) are the principal values of following trigonometric equations ( sin theta=-frac{1}{sqrt{2}} )
2) ( cos theta=-frac{sqrt{3}}{2} )
3) ( tan theta=sqrt{3}-2 )
A. ( p_{1}<p_{2}<p_{3} )
в. ( p_{1}<p_{3}<p_{2} )
c. ( p_{3}<p_{1}<p_{2} )
D . ( p_{2}<p_{3}<p_{1} )
11
1366If tan 0 + sec 0 = 1.5, find sin e, tan , and
Illustration 2.6
sec e.
11
1367Prove that
( frac{sin A+sin B}{cos A+cos B}=tan frac{A+B}{2} )
11
1368Solve: ( |cos x|=cos x-2 sin x )
A ( cdot x=(2 n+1) pi+frac{pi}{4} )
B ( cdot x=(2 n+1) pi+frac{pi}{2} )
C ( cdot x=2 n pi+frac{pi}{4} )
D ( cdot x=2 n pi+frac{pi}{2} )
11
1369If ( (2 cos x+sin x)=1, ) then sum of all
possible value of ( (7 cos x+6 sin x) )
is
11
137015. If sin x + sin y 2 cos a cos x ve R, then sin y + cos a is
equal to
11
1371The general solution of the equation
( sin x+cos x=1 ) is
A ( cdot x=2 n pi+frac{pi}{2}, n=0,pm 1,pm 2 )
B ( cdot x=n pi+left((-1)^{n}+1right) frac{pi}{4}, n=0,pm 1,pm 2 )
C ( x=n pi+left((-1)^{n}-1right) frac{pi}{4}, n=0,pm 1,pm 2 )
D. ( x=2 n pi, n=0,pm 1,pm 2 )
11
1372If ( A+B+C=pi ) then the expression ( frac{sin 2 A+sin 2 B-sin 2 C}{sin 2 A+sin 2 B+sin 2 C} ) reduces to11
137311 UUD 100
8. Prove that tan a + 2 tan 20 + 4 tan 4a + 8 cot 8a=cot a.
(IIT-JEE 1988)
11
1374Illustration 4.15 Find the number of solutions of [cos x] +
sin x = 1 in a <x<37 (where [.] denotes the greatest integer
function).
11
1375The number of values of x in the interval [0, 511] satisfying
the equation 3 sinx-7 sin x +2=0 is (1998 – 2 Marks)
(a) 0 (b) 5
(c) 6
(d) 10 |
11
1376sin 0 + cos 0
69. If
ue of sin”e – cose is
sin 0 –coso = 3, then the val-
11
137724. If cot (0 – 0), 3 coté, cot (0+ a) are in A.P. and is not
4 sine
an integral multiple of 5, then the value of 4S
3sinº a
11
137823. The general solution of the equation 8 cos x cos 2x cos 4x
= sin 6x/sin x is
a. x = (n/7) + (1/21), ne z
b. x = (21/7) + (10/14), Vnez
c. x = (n/7) + (1/14), Vnez
d. x = (n )+ (1/14), V ne z
11
1379If ( frac{a x}{cos theta}+frac{b y}{sin theta}=a^{2}-b^{2} ) and
( frac{a x sin theta}{cos ^{2} theta}-frac{b y cos theta}{sin ^{2} theta}=0 ) then ( (a x)^{2 / 3}+ )
( (b y)^{2 / 3}=left(a^{2}-b^{2}right)^{2 / 3} ? )
A. True
B. False
11
1380If ( sin a=frac{3}{7} ) and ( cos a<0, ) what is the
value of ( tan a )
A ( cdot frac{-(3 sqrt{20})}{10} )
B. ( frac{-(sqrt{10})}{10} )
c. ( frac{-(3 sqrt{10})}{20} )
D. ( frac{(2 sqrt{5})}{10} )
11
1381Illustration 3.96
5x +12y+ 7xy
If x2 + 12 = x2,2 then find the range of
11
1382The terminal arm is in II quadrant, what
are the measures of possible angles?
A . In between ( 90^{circ} ) and ( 180^{circ} ) or ( -270^{circ} ) and ( -180^{circ} )
B. In between ( 180^{circ} ) and ( 270^{circ} ) or ( -90^{circ} ) and ( -180^{circ} )
C. In between ( -90^{circ} ) and ( 0^{circ} ) or ( 270^{circ} ) and ( 360^{circ} )
D. None of these
11
1383The value of ( sin 75^{circ}= )
( ^{A} cdot frac{2-sqrt{3}}{sqrt{2}} )
в. ( frac{sqrt{3}+1}{2 sqrt{2}} )
c. ( frac{sqrt{3}-1}{2 sqrt{2}} )
D. ( frac{sqrt{3}+1}{sqrt{2}} )
11
1384Solve the equations
( 3left(sec ^{2} theta+tan ^{2} thetaright)=5 )
11
1385If the angle ( alpha ) lies in the first quadrant
and ( tan alpha+cot alpha=2, ) then the value
of ( sqrt{tan alpha}+sqrt{cot alpha} ) is
A . -4
B. – –
( c cdot 2 )
D. 4
11
13863.
Let n be an odd integer. If sin no = Eb sin’
r=0
value of 0, then
(a) b=1, b, = 3
(c) b. =-1, b =n
an odd integer. If sin ne = { b sin” e, for every
(1998 – 2 Marks)
(b) b =0,b=n
(d) b = 0, b, = n2 – 3n+3
11
1387( tan 9-tan 27-tan 63+tan 81= )11
138866. The total number of solutions of log sin x| = -x + 2x in
[0, 1] is equal to
a. 1
b. 2
c. 4
d. none of these
11
1389Solve:
( frac{tan theta}{sec theta+1}+frac{tan theta}{sec theta-1}=2 csc theta )
11
139063. If sino + coseco = 2, then val-
ue of sin 1000 + cosec1009 is
equal to :
(1) 1
(2) 2
(3) 3
(4) 100
11
1391If ( 8 tan A=15, ) then the value of ( frac{sin A-cos A}{sin A+cos A} ) is:
A ( cdot frac{7}{23} )
в. ( frac{11}{23} )
c. ( frac{13}{23} )
D. ( frac{17}{23} )
11
1392[
begin{aligned}
operatorname{Let} A &=left{theta: 2 cos ^{2} theta+sin theta leq 2right} text { and } \
B &={theta: pi / 2 leq theta leq 3 pi / 2} . text { Then }
end{aligned}
]
find the value of ( boldsymbol{A} cap boldsymbol{B} )
11
13939. If sin x + cosec x = 2, then sin”x + cosec”x is equal to
a. 2
b. 21
c. 2n-1
d. 2n-2
11
1394( (sin theta+cos theta)(1-sin theta cos theta) ) can be
written as:
( mathbf{A} cdot sin theta+cos theta )
( mathbf{B} cdot sin ^{3} theta-cos ^{3} theta )
( mathbf{c} cdot sin ^{3} theta+cos ^{3} theta )
( mathbf{D} cdot sin theta-cos theta )
11
1395Assertion ( (A): sin ^{2} theta+sin ^{2}left(theta+60^{0}right)+ )
( sin ^{2}left(theta-60^{0}right)=frac{3}{2} )
Reason ( (mathrm{R}): cos alpha+cos left(120^{0}+alpharight)+ )
( cos left(120^{0}-alpharight)=0 )
A. Both A and R are true and R is the correct explanation to A
B. Both A & R are true but R is not the correct explanation to ( A )
c. A is true, R is false
D. A is false, R is true
11
1396Sin 2B
85. If tan(a-B) =
• If tanſa-B) –
Cos 2R, then
a. tan a=2 tan B
c. 2 tan a=3 tan ß
b. tan B=2 tan a
d. 3 tan a= 2 tan ß
11
1397360° 540°
56. cosec – + cosec
=
7
1800
90°
a. cosec
b. cosec
– 2001
180°
c. sec
900
d. sec
11
1398If ( cos theta=-frac{3}{5}, pi^{c}<theta<frac{3 pi^{c}}{2}, ) find the
value of ( frac{csc theta+cot theta}{sec theta-tan theta} )
11
1399If ( cos A=frac{4}{5}, cos B=frac{12}{13}, frac{3 pi}{2}< )
( A, B<2 pi, ) find the values of the
following.
( (mathrm{i}) cos (boldsymbol{A}+boldsymbol{B}) )
(ii) ( sin (A-B) )
11
14006. Find the smallest positive root of the equation
sin(1 – x) = cos x .
11
1401If ( boldsymbol{theta}+boldsymbol{phi}=frac{boldsymbol{pi}}{boldsymbol{4}}, ) then ( (mathbf{1}+tan boldsymbol{theta})(mathbf{1}+ )
( tan phi) ) is equal to
A .
B. 2
( c cdot frac{5}{2} )
D.
11
1402Solve: ( 4 cos ^{2} x+6 sin ^{2} x=5 )
A ( cdot x=n pi+frac{pi}{2} )
B. ( x=n pi-frac{pi}{3} )
c. ( x=n pi pm frac{pi}{4} )
D. None of these
11
1403What is the value of ( sin ^{2} 25^{0}+sin ^{2} 65^{0} ? )11
1404The equation ( sec ^{2} theta=frac{4 x y}{(x+y)^{2}} ) is only
possible, when
( mathbf{A} cdot x=y )
в. ( xy )
D. None of these
11
1405If both the distinct roots of the equation
( |sin x|^{2}+|sin x|+b=0 ) in ( [0, pi] ) are
real, then the value of ( b ) is
( mathbf{A} cdot[-2,0] )
B . (-2,0)
( c cdot[-2,0) )
D. None of these
11
1406Simplify, using trigonometric tables
( tan 63^{circ} 12^{prime}-cos 12^{circ} 42^{prime} )
11
140742. The value of k if the equation 2 cos x + cos 2kx = 3 has
only one solution is
b. 2
d. 1/2
a. O
c. 12
11
1408If ( cos (A-B)=frac{3}{5} & tan A tan B=2 )
then
This question has multiple correct options
A ( cdot cos A cos B=frac{1}{5} )
B. ( sin A sin B=-frac{2}{5} )
( c cdot cos (A+B)=-frac{1}{5} )
D. ( sin A sin B=frac{2}{5} )
11
14091
43. Number of solution(s) satisfying the equation
sin x
sin 4x in [0, 47] equals
sin 2x
a. 0
c. 4
b. 2
d. 6
11
1410Prove that:
[
begin{array}{c}
frac{tan theta-cot theta}{sin theta cos theta}=sec ^{2} theta-operatorname{cosec}^{2} theta= \
tan ^{2} theta-cot ^{2} theta
end{array}
]
11
1411Prove that ( frac{sin theta-cos theta+1}{sin theta+cos theta-1}= )
( frac{1}{sec theta-tan theta}, ) using the identity
( sec ^{2} theta=1+tan ^{2} theta )
11
1412Show that ( tan ^{2} theta-frac{1}{cos ^{2} theta}=-1 )11
1413Prove that ( : sqrt{left(sec ^{2} Theta+operatorname{cosec}^{2}right.}= )
( tan Theta+cot Theta )
11
1414Express in radians ( 345^{circ} 25^{prime} 36^{prime prime} )
A . ( 1.726268 pi^{c} )
B . ( 1.3465338 pi^{c} )
( mathbf{c} cdot 1.91903 pi^{c} )
D. ( 1.1978258 pi^{c} )
11
14153.
Which one is not periodic
(a) sin3x|+sinºx
(c) cos 4x + tan²x
[2002]
(b) cos Vx + cos x
(d) cos2x + sinx
11
1416In a circle diameter, ( 40 mathrm{cm} ), the length of a chord is ( 20 mathrm{cm} . ) Find the length of
minor arc of the chord.
11
1417Evaluate
( 8 sqrt{3} operatorname{cosec}^{2} 30^{circ} sin 60^{circ} cos 60^{circ} cos ^{2} 45^{circ} )
A ( cdot 2 sqrt{3} )
B. ( 4 sqrt{3} )
( c cdot 8 sqrt{3} )
D. ( 12 sqrt{3} )
11
1418f ( operatorname{cosec} A=2, ) find the value of
( frac{1}{tan A}+frac{sin A}{1+cos A} )
11
14195. Number of solutions of the equation
(√3+ 1)² + (13-1 * = 23r is
11
14209.
Find the values of xe(-1, + 7) which satisfy the equation
8(1+cos xl+ cos2xl+cosº xlt…) – 43 (1984 – 2 Marks)
11
1421Which one of the following is correct?
( mathbf{A} cdot sin 45^{0} cos 45^{0}=1 )
B ( cdot sin ^{2} 45^{0}-cos ^{2} 45^{0}=1 )
( mathbf{C} cdot sin 30^{0}+cos 60^{0}=1 )
D ( cdot cos ^{2} 30^{0}-cos ^{2} 60^{0}=1 )
11
142283. If both the distinct roots of the equation (sin xß+ |sin x! +
b = 0 in [0, 1] are real, then the values of b are
a. [-2, 0]
b. (-2,0)
c. [-2,0)
d. none of these
11
1423Find the domain of definition of the
following function:
( y=sqrt{sin ^{2} x-sin x} )
11
1424If ( boldsymbol{m} sin boldsymbol{theta}=boldsymbol{n} sin (boldsymbol{theta}+boldsymbol{2} boldsymbol{alpha}), ) then
( tan (boldsymbol{theta}+boldsymbol{alpha}) cdot cot boldsymbol{alpha} ) equal to
A. ( frac{1-n}{1+n} )
в. ( frac{m+n}{m-n} )
c. ( frac{m-n}{m+n} )
D. ( frac{1+n}{1-n} )
11
14253. Which of the following quantities are rational?11
1426Illustration 4.10 If 2tan” x – 5sec x = 1 for exactly seven
distinct values of x € [0, nm/2], n e N then find the greatest
value of n.
11
142778. In triangle ABC, if angle C is 90° and the area of triangle
is 30 sq. units, then the minimum possible value of the
hypotenuse c is equal to
a. 3072
b. 6012
c. 12072
d. 2730
11
1428If an angle ( a ) is divided into two parts ( A )
& ( B ) such
that ( A-B=x ) and ( tan A: tan B= )
( K: 1, ) then the value of ( sin x ) is
A ( cdot frac{k+1}{k-1} sin alpha )
B. ( frac{k}{k+1} sin alpha )
c. ( frac{k-1}{k+1} sin alpha )
D. none of the above
11
1429Solve:
( frac{cos 70}{sin 20}+cos 59 csc 31 )
11
14307. Let 0<,<0,<0z< … denote the positive solution of the
equation 3 + 3 cos 0 = 2 sin? . The value of 03 + 0, is
a. 61
b. 71
c. 87
d. 411
11
143164. Ifre ( 7, 37), then 4 cos ( 7
) + VasinⓇx+ sin? 2x
is always equal to
a. 1
C.-
2
y equal cos
b. 2
d. none of these
Pone of these
020
11
14323. Find the values of x € (-, 1) which satisfy the equation
8 (1+cos xl+cos xl+cos” x + …) = 43
(IIT-JEE 1984)
11
1433Assertion
The system of linear equations ( boldsymbol{x}+(sin boldsymbol{alpha}) boldsymbol{y}+(cos boldsymbol{alpha}) boldsymbol{z}=mathbf{0} )
( x+(cos alpha) y+(sin alpha) z=0 quad ) has a non
( -boldsymbol{x}+(sin boldsymbol{alpha}) boldsymbol{y}-(cos boldsymbol{alpha}) boldsymbol{z}=mathbf{0} )
trivial solution for only one value of ( boldsymbol{alpha} )
Iying between 0 and ( pi )
Reason ( left|begin{array}{ccc}sin x & cos x & cos x \ cos x & sin x & cos x \ cos x & cos x & sin xend{array}right|=0 ) has no
solution in the interval ( -frac{pi}{4}<x<frac{pi}{4} )
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
11
1434Find principal and general solution of the equation, ( cot x=-sqrt{3} )11
1435The number of distinct solutions of
( sin 5 theta cdot cos 3 theta=sin 9 theta cdot cos 7 theta ) in ( theta in )
( [mathbf{0}, boldsymbol{pi} / mathbf{2}] ) is
( A cdot 4 )
B. 5
( c cdot 8 )
D.
11
143675. Given that a, b, c are the sides of a AABC which is right
angled at C, then the minimum value of -+-
a. 0
c. 6
b. 4
d. 8
200
11
1437Prove that ( frac{cos (pi+x) cos (-x)}{sin (pi-x) cos left[frac{pi}{2}+xright]}= )
( cot ^{2} x )
11
143855. The total number of solutions of cosx = V1-sin 2x in
[0, 21) is equal to
b. 3
c. 5
d. none of these
a. 2
11
1439( frac{1-sin A}{cos A} ) is equal to
( A cdot frac{cos A}{1+sin A} )
B. ( frac{sin A}{1-cos A} )
c. ( frac{tan A}{1+tan A} )
D. ( frac{tan A}{1+cos A} )
11
1440Illustration 4.25
Solve cos 0+ cos 30 – 2 cos 20= 0.
11
1441A minimum value of ( sin x cos 2 x ) is-
( A cdot 1 )
B. – 1
c. ( -2 / 3 sqrt{6} )
D. None of these
11
1442Illustration 3.86 Prove that
cos 20° cos 40° cos 60° cos 80º = 1/16.
11
1443The total number of solutions of
( cos x=sqrt{1-sin 2 x} ) in ( [0,2 pi] ) is equal
to
A . 2
B. 3
c. 5
D. None of these
11
1444Find the number of solutions of the
equations;
( mathbf{2}^{cos x}=|sin x| ) when ( boldsymbol{x} epsilon[-2 pi, 2 pi] )
11
144581. The equation cos*x + b cos x + 1 = 0 will have a solution
if b belongs to
a. (-,2]
b. [2,00)
c. (-, -2]
d. none of these
11
1446tv
11. If Ois eliminated from the equations x = a cos(-a) and
ad y = b cos(0-3), then
me to say
(a-B) is equal to
a b² ab
a. sec (a – B) b. cosec°(a – b)
c. cos?( – B) d. sin?(a-B)
COS (a – B
11
1447Find the general solution of ( x cos ^{2} 2 x+ )
( cos ^{2} 3 x=1 )
A ( cdot(2 k+1) frac{pi}{10}, k in I )
В ( cdot(pi+1) frac{pi}{10} ; k in I )
( c cdot(2 k-1) frac{pi}{10}, k in I )
D. Both (A) and (C)
11
1448Illustration 4.33
Solve 2 tan 0-cot =-1.
11
144946. If 3 tan(0-159) = tan(@+ 15º), then 0 is equal to (ne z
a. nit +
b. nt +
+
c. nt +
d. none of these
11
1450Which one of the following is the value of ( cos 170^{0} cos 10^{0}-sin 170^{0} sin 10^{0} ? )
A . -2
B. – 1
( c . )
D.
11
1451If ( 2 sin 2 theta=sqrt{3}, ) then ( tan theta ) is
( A )
B. ( frac{1}{sqrt{3}} )
( c cdot sqrt{3} )
D. ( frac{1}{sqrt{2}} )
11
1452Find the area of the isosceles triangle
with base ( 16 mathrm{cm} ) and vertical angle
( mathbf{6 0}^{circ} mathbf{4 0}^{prime} )
11
145316. If (sin a) x2 – 2x + b 2 for all the real values of x = 1
and a e (0, T/2) U (F/2, T), then the possible real values
of b is/are
a. 2
b. 3
c. 4
d. 5
11
145431. Range of f(0) = cos? O (cos2 0 + 1) + 2 sin? O is
a. [3/4, 1]
b. [3/16, 1]
c. [3/4, 7/4]
d. [7/4, 2]
11
1455Find the ratio of ( sin x, cos x, tan x, ) where
( mathbf{x}=mathbf{4 5}^{mathbf{0}} ? )
11
1456( A B C ) is a triangle in which ( A B= )
( A C=4 mathrm{cm} ) and ( angle A=90^{circ} . ) Calculate
the length of perpendicular from ( boldsymbol{A} ) to ( boldsymbol{B C} )
A. ( 2.83 mathrm{cm} )
B. ( 1.414 mathrm{cm} )
( c .2 .6 mathrm{cm} )
D. 2.20 cm
11
1457Show that ( frac{sin 2 alpha+sin 2 beta}{cos 2 alpha-cos beta}=cot (beta- )
( boldsymbol{alpha} )
11
1458Illustration 2.59
Show that tan 1° tan 2° … tan 89° = 1.
11
1459If ( cos ^{2} x+cos ^{2} 2 x+cos ^{2} 3 x=1 ) then
A ( cdot x=(2 n+1) frac{pi}{4}, n epsilon I )
B. ( x=(4 n+1) frac{pi}{4}, n in I )
c. ( x=n pi frac{pi}{4}, n epsilon I )
D. none of these
11
1460The incorrect statement is
( mathbf{A} cdot sin theta=-frac{1}{5} )
( mathbf{B} cdot cos theta=1 )
( mathbf{c} cdot sec theta=frac{1}{2} )
( mathbf{D} cdot tan theta=20 )
11
14612 TV
Illustration 3.71
Find the value of cos
6T
+cos — +cos —
11
1462Illustration 4.54
Solve 1 + sinx sin? – =0.
11
1463Given that ( cos 50^{circ} 18^{prime}= )
0.6388 and ( cos 50^{circ} 42^{prime}=0.6334, ) then
the possible value of ( cos 50^{circ} 20^{prime} ) is
A . 0.6293
B. 0.6307
c. 0.636
D. 0.6414
11
1464Prove that: ( cos ^{2} boldsymbol{alpha}+cos ^{2}(boldsymbol{alpha}+boldsymbol{beta})- )
( boldsymbol{2} cos boldsymbol{alpha} cos boldsymbol{beta} cos (boldsymbol{alpha}+boldsymbol{beta})=sin ^{2} boldsymbol{beta} )
11
1465( 110^{circ} 30^{prime} ) in radians is :
A ( cdot frac{221 pi^{c}}{360} )
в. ( frac{225 pi^{c}}{360} )
c. ( frac{231 pi^{c}}{360} )
D. ( frac{229 pi^{c}}{360} )
11
1466Equation ( 6 sin ^{2} theta-5 sin theta+1=0 ) is
satisfied by
A ( cdot theta=frac{pi}{2} )
B. ( theta=frac{pi}{3} )
( mathbf{c} cdot theta=frac{pi}{4} )
D・ ( theta=frac{pi}{6} )
11
1467Find which of the number of the form
( left(n pi-t a n^{-1} 3right), ) where ( n epsilon l, ) are
solution for ( 12 tan 2 x+frac{sqrt{10}}{cos x}+1=0 )
( mathbf{A} cdot n=k, k epsilon z )
B. ( n=(3 k+1), k epsilon z )
c. ( n=(2 k+1), k epsilon z )
D. ( n=(4 k+1), k epsilon z )
11
1468Illustration 2.60
Find the value of cos? *+ cos20
16
16
2777
+ cos-
CO2 In
16
+ cos2
16
11
146917. Find the number of solutions of 8 € [0, 21] satisfying the
equation (log/z tan e) (Vlogan o 3 + log/5 33) — 1.
11
147054. The equation sinfx + cos x + sin 2x + a=0 is solvable for
a. – 5/2 sas 1/2 b. – 3 sasi
c. – 3/2 s as 1/2 d. – 1 sasi
11
1471( cos x+cos y=frac{4}{5}, cos x-cos y=frac{2}{7} )
The value of ( 14 tan left(frac{x-y}{2}right)+ )
( mathbf{5} cot left(frac{boldsymbol{x}+boldsymbol{y}}{mathbf{2}}right) ) is
( mathbf{A} cdot mathbf{0} )
B.
( c cdot frac{5}{4} )
D.
11
1472Which of the following is correct?
( A cdot sin 1^{circ}>sin 1 )
B. ( sin 1^{circ}<sin 1 )
( mathbf{c} cdot sin 1^{circ}=sin 1 )
D. ( sin 1^{circ}=frac{pi}{180} sin 1 )
11
1473Suppose the point with coordinates (-12,5) is on the terminal side of angle
( theta, ) the value of the sine trigonometric function of ( theta . ) is ( frac{a}{13} ) Find ( a )
11
1474Solve ( frac{sin 30^{circ}+tan 45^{circ}-cos e s 60^{circ}}{sec 30^{circ}+cos 60^{circ}+cos 45^{circ}} )11
1475Illustration 3.85 The product of the sines of the angles of a
triangle is p and the product of their cosines is q. Show that
the tangents of the angles are the roots of the equation qx –
px + (1 + q) x-p= 0.
11
1476The value of the expression 13 cosec 20°-sec 20° is
equal to
(1988-2 Marks)
(a) 2
(b) 2 sin 20%sin 40°
(c) 4
(d) 4 sin 20%sin 40°
11
1477If ( tan theta=frac{x sin phi}{1-x cos phi}, tan phi= )
( frac{y sin theta}{1-y cos theta}, ) then ( frac{x}{y}= )
A. ( frac{sin phi}{sin theta} )
B. ( sin theta )
sin ( phi )
c. ( frac{1-cos phi}{1-cos theta} )
D. ( frac{1-cos theta}{1-cos phi} )
11
1478Solve ( tan theta=2, ) then ( theta=n pi+ )
( a, ) where ( a=tan ^{-1}(2), n in I )
If true then enter 1 and if false then
enter 0
11
1479Solve
( 7 sin ^{2} theta+3 cos ^{2} theta=4 )
11
1480TT
37
3. The value of (1 + com 5 ) (+ cos?(1 + cos *)
3. The value of
+ cos-
+ cos
+ COS
210
a. 1/4
c. 1/8
b. 3/4
d. 3/8
(IIT-JEE 1984)
11
1481Find the value of other five
trigonometric ratios:
( sec x=frac{13}{5}, x ) lies in fourth quadrant
11
1482If ( sec theta=1 ; 0 leq theta<12^{circ}, ) then the value
of ( boldsymbol{theta} ) is
A ( .5^{circ} )
B . 0
( c cdot 1^{c} )
D. ( 2^{circ} )
11
1483ii.

Illustration 3.40 Prove that
sin20
sin 20
= tan
– = coto
1 + cos2e
1- cos 20
1 + sin 20 + cos20
1 + sino – cos o
== tan 0/2
1+ sin 20 – cos 20
1+ sin 0 + cose
cos 20
– = tan (T/4 – ) vị.
= tan
1 + sin 20
1 + sine
= cot Oiv.
cos e
A12
11
1484[2012]
24. The equation esinx – e-sinx_4=0 has :
(a) infinite number of real roots
(b) no real roots
© exactly one real root
(d) exactly four real roots
11
1485If ( operatorname{cosec} x-c o t x=frac{1}{3}, ) where ( x neq 0 )
then the value of ( cos ^{2} x-sin ^{2} x ) is
A ( cdot frac{16}{25} )
в. ( frac{9}{25} )
( c cdot frac{8}{25} )
D. ( frac{7}{25} )
11
1486If ( alpha, beta ) are two different values of ( theta ) lying
between 0 and ( 2 pi ) which satisfy the
equation ( 6 cos theta+8 sin theta=9 . ) Find
( sin (alpha+beta) )
11
1487If ( A=tan 6^{0} tan 42^{0} ) and ( B= )
( cot 66^{0} cot 78^{0}, ) then
( A cdot A=2 B )
B. A=1/3 B
( c cdot A=B )
D. 3A=2B
11
1488If ( tan alpha=2 sqrt{2}, ) then the value of
( frac{tan alpha}{frac{sin ^{3} alpha}{cos alpha}+sin alpha cdot cos alpha} )
( mathbf{A} cdot mathbf{0} )
B. 2
c. ( 2 sqrt{2} )
( D )
11
1489If ( cos B cos C+sin B sin C sin ^{2} A=1 )
then triangle ( A B C ) is
A. isosceles and right angled
B. equilateral
C. isosceles whose equal angles are greater than ( pi / 4 )
D. none
11
1490The value of ( left(1+tan ^{2} thetaright) sin ^{2} theta ) is
( mathbf{A} cdot sin ^{2} theta )
B. ( cos ^{2} theta )
( mathbf{c} cdot tan ^{2} theta )
( D cdot cot ^{2} theta )
11
1491( sqrt{3} sin theta=cos theta, ) find the value of
( frac{3 cos theta+2 sin theta}{2 cos ^{2} theta-2} )
11
149218. 1-r cos a
1-r cos O is equal to
r sin a
a. tan 20
c. sin 20
b. cot 20
d. cos 20
11
14935. For 0 <$<
, the solution(s) of
È cosec (0+ (m = 1, cosec( 6 + mut) = 472
4
m=1
is (are)
a. 7/4
c. 7/12
b. 7/6
d. 57/12
(IIT-JEE 2009)
11
1494In ( Delta A B C, 4 sin A cos B=1 ) and
( tan B=3 tan A . ) Then ( sin C= )
A ( cdot frac{1}{4} )
в. ( frac{1}{2} )
( c cdot frac{3}{4} )
D.
11
1495If A lies in the second quadrant and ( mathbf{3} tan boldsymbol{A}+mathbf{4}=mathbf{0}, ) the value of ( mathbf{2} cot boldsymbol{A}- )
( 5 cos A+sin A ) is equal to
A ( cdot-frac{53}{10} )
в. ( frac{23}{10} )
( c cdot-frac{37}{10} )
D. ( frac{7}{10} )
11
149677. The value of tan 6° tan 42° tan 66° tan 78° is
a. 1
b. 1/2
c. 1/4
d. 1/8
11
1497Illustration 2.43 For real values of e, which of the following
is/are always positive?
a. cos(cos )
b. cos(sin )
:
c. sin(cos )
d. sin(sin )
11
1498In A ABC, prove that cos A + cosB +
Illustration 3.100
cos C < 3/2.
11
1499A wheel makes 360 revolutions in one
minute. Through how many radians does it turn in one second?
11
1500Show that
( (sec theta-tan theta)^{2}=frac{1-sin theta}{1+sin theta} )
11
1501Illustration 2.46
Find the range of f(x) = cos²x + sec?x.
11
150272. The value of tan 75° is:
13+1 13-1
(1) T3 – 1
(2) 73+1
1-13 1+13
(3) 1+13 (4) 1-13
11
150312. Let o, o e [0, 21t] be such that 2 cose (1 – sin o) = sin e
(tang+cot)cose1, tan (272-e) >0 and
-1 <sine <-
, then o cannot satisfy
(2012)
m) o<«<
<<
11
1504If ( tan theta_{1}=k cot theta_{2}, ) then ( frac{cos left(theta_{1}+theta_{2}right)}{cos left(theta_{1}-theta_{2}right)}= )
A. ( frac{1+k}{1-k} )
B ( cdot frac{1-k}{1+k} )
c. ( frac{k+1}{k-1} )
D. ( frac{k-1}{k+1} )
11
1505tan(In 6) tan(In 2) tan(In 3)
23. If –
tan(In 6) – tan(In 2) – tan(In 3)
= k, then the value of k
is
11
150660. If cos x = tan y, cos y = tan z, cos z = tan x, then the value
of sinx is
a. 2 cos 18°
b. cos 18°
c. sin 18°
d. 2 sin 18°
11
1507If ( boldsymbol{x}+boldsymbol{y}=frac{boldsymbol{pi}}{2}, ) then prove that ( cos (boldsymbol{x}+ )
( boldsymbol{y})=mathbf{0} )
11
1508( cos 60^{0} times cos 30^{0}-sin 60^{0} times sin 30^{0}=0 )11

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