# Some Applications Of Trigonometry Questions

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

#### List of some applications of trigonometry Questions

Question NoQuestionsClass
1A ladder of length ( 20 m ) is resting
against a wall ( 10 m ) high such that the
top of the ladder touches the top edge of the wall. Find the distance of the foot of the ladder from the foot of the wall?
A ( cdot frac{10}{sqrt{3}} m )
в. ( 10 m )
c. ( 10 sqrt{3} mathrm{m} )
.
D. 30 ( m )
10
2( ln ) a ( Delta A B C, ) if ( angle B=60^{circ}, ) prove that
( (a+b+c)(a-b+c)=3 c a )
10
3Two boats approach a light house in mid-sea from opposite directions. The angles of elevation of the light house
from two boats are ( 30^{circ} ) and ( 45^{circ} )
respectively. If the distance between two boats is ( 100 mathrm{m}, ) find the height of the light house.
10
451.
If the angle of elevation of the
sun changes from 45° to 60°, then
the length of the shadow of a
pillar decreases by 10 m. The
height of the pillar is :
(1) 5(3-13) metre
(2) 5(13 + 1) metre
(3) 15 (13 + 1) metre
(4) 5(3 + 13) metre
10
5Illustration 3.18 The length of the shadow of a vertical
pole of height h, thrown by the sun’s rays at three different
moments are h, 2h and 3h. Find the sum of the angles of
elevation of the rays at these three moments.
10
666. The length of the shadow of a
tower is 9 metres when the sun’s
altitude is 30°. What is the height
of the tower?
(1) 3.73 m
(2) 4
m
(3) 9.73 m
(4 90
m
10
7A vertical tower stands on a horizontal
plane and is surmounted by a vertical flagstaff of height 5 meters. At a point
on the plane, the angles of elevation of the bottom and the top of the flagstaff
are respectively ( 30^{circ} ) and ( 60^{circ} . ) Find the
height of the tower.
10
8The shadow of a tower standing on a
level plane is found to be ( 50 m ) longer
when sun’s elevation is ( 30^{circ} ) than when
it is ( 60^{circ} . ) Find the height of the tower.
10
9A vertical tree stands on a hill side that
makes an angle ( alpha ) with the horizontal From a point directly up the hill from the tree, the angle of elevation of three top
is
( beta . ) From a point ( mathrm{m} mathrm{cm}, ) further up the
hill, the angle of depression of the tree top is ( gamma ). If the tree is ( mathrm{H} mathrm{cm} ). tall, express
( mathrm{H} ) in terms of ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} )
10
10True or False
A person stands at a point A due south of a tower and observes his elevation is
( 60^{circ} . ) He then walks westwards towards
where the elevation is ( 45^{circ} . ) At a point ( C )
on ( A B ) produced he finds it to be ( 30^{circ} )
then ( A B=2 B C )
A. True
B. False
10
11A boy playing on the roof top of a ( 10 mathrm{m} ) high building throws a ball with a speed
of ( 10 mathrm{m} / mathrm{s} ) at an angle of ( 30_{0} ) with the
horizontal. How far from the throwing point will the ball be at the height of 10m from the ground.
( sin 30^{0}=frac{1}{2} )
( cos 30^{0}=frac{sqrt{3}}{2} )
A. 8.66 ( mathrm{m} )
B. 5.20 ( mathrm{m} )
c. 4.33 m
D. 2.60 ( mathrm{m} )
10
12Illustration 2.18
A man observes when he has climbed up
1

he length of an inclined ladder, placed against a wall,
the angular depression of an object on the floor is a. When he
climbs the ladder completely, the angle of depression is ß. If
the inclination of the ladder to the floor is 0, then prove that
e_3cot ß-cot a
10
13A circus is climbing a ( 20 m ) long rope, which is tightly stretched and tied from the top of vertical pole to the ground. Find the height of the pole if the angle made by the rope with the ground level
is ( 30^{circ} )
A ( .24 m )
в. 18 т
c. ( 28 m )
D. ( 10 m )
10
1467. A man from the top of a 100
metre high tower sees a car mov-
ing towards the tower at an an-
gle of depression of 30°. After
some time, the angle of depres-
sion becomes 60°. The distance
(in metres) travelled by the car
during this time is
(1) 100 13
2003
(2)
(3)
100/3
vs
(4)20073
10
15Find the angle of evelation of the sun
when the shadow of a pole ( h ) meter high is ( sqrt{3} h ) meters long
10
1661. A circus artist is climbing from
the ground along a rope
stretched from the top of a ver-
tical pole and tied at the
ground. The height of the pole
is 12 m and the angle made
by the rope with ground level
is 30°. The distance covered by
the artist in climbing to the
top of pole is
(1) 12 /2 m. (2) 12 m.
(3) 24 m. (4) 18 m.
10
1774. If the angle of elevation of a
cloud from a point h metres
above a lake is B and the angle
of depression of its reflection
in the lake is a, then the
height of the cloud is:
h cosec (a + B)
(1) cosec (a + B)
(2) h cosec (a + b) sin (a – B)
(3) h sin (a + b) cosec (a – B)
h cosec (a + B)
(4) sin(a – b)
10
18A pole ( 15 mathrm{m} ) long rests against a vertical
wall at an angle of ( 30^{circ} ) with the ground
How high up the wall does the pole reach?
( A cdot 5 m )
в. 7
c. ( 7.5 mathrm{m} )
( D cdot 8 m )
10
1975. If the angle of elevation of the Sun
changes from 30° to 45°. the
length of the shadow of a pillar
decreases by 20 metres. The
height of the pillar is
(1) 20 (3-1) m
(2) 20 (13 + 1) m
(3) 10 (3-1) m
(4) 10 (213 +1) m
10
2087. A person of height 2m wants
get a fruit which is on a pole
height (
m. If he stands at a
distance of
m from the foot
of the pole, then the angle at
which he should throw the stone.
so that it hits the fruit is
(1) 60° (2) 45°
(3) 90°
(4) 30°
10
21( A B C ) is a triangular park with ( A B= ) ( A C=100 ) metres. A vertical tower is
situated at the mid-point of ( B C . ) If the
angles of elevation of the top of the tower at ( A ) and ( B ) are ( cos ^{-1}(3 sqrt{2}) ) and ( operatorname{cosec}^{-1}(2 sqrt{2}) ) respectively, then the height of the tower (in metres is):
A. ( 10 sqrt{5} )
the ( 6 sqrt{55} )
в. ( frac{100}{3 sqrt{3}} )
c. 20
D. 25
10
2268. The distance between two ver-
tical poles is 60 m. The height
of one of the poles is double
the height of the other. The
angles of elevation of the top
of the poles from the middle
point of the line segment join-
ing their feet are complemen-
tary to each other. The heights
of the poles are :
(1) 10 m and 20 m
(2) 20 m and 40 m
(3) 20.9 m and 41.8 m
(4) 15.12 m and 30./2m
10
2352. If the height of a pole is 23
metre and the length of its
shadow is 2 metre, then the
angle of elevation of the sun
is
(1) 90°
(3) 30
(2) 45°
(4) 60°
10
2475. The longer side of a parallelogram
is 10 cm and the shorter 6 cm. If
the longer diagonal makes an
angle 30° with the longer side,
find the length of the longer di-
agonal.
(1) (513 + VTT)em
(2) (573 +7cm
(3) (3/5 +11)cm
10
25A vertical to stands on a horizontal
plane and is surmounted by a vertical flag staff of height 6 meters. At point 0 n the ( p ) angle of elevation of the bottom and the top of the flag staff are respectively ( 30^{prime} ) and ( 60^{*} ). Find the height of tower-
A. ( 5 mathrm{m} )
B. 2.5m
( c cdot 10 m )
D. 7.5m
10
2627 A bird flies in a circle on a horizontal plane. An observer
stands at a point on the ground. Suppose 60° and 30° are
the maximum and the minimum angles of elevation of the
bird and that they occur when the bird is at the points P and
Orespectively on its path. Let O be the angle of elevation of
the bird when it is a point on the arc of the circle exactly
midway between P and Q. Find the numerical value of
tan-0.(Assume that the observer is not inside the vertical
projection of the path of the bird.) (1998 -8 Marks)
10
27Choose the correct answer from the
alternatives given A tree is broken by the wind. If the top of the tree struck the ground at an angle of
( 30^{circ} ) and at a distance of ( 30 mathrm{m} ) from the
root, then the height of the tree is
B. ( 20 sqrt{3} m )
( mathbf{c} cdot 25 sqrt{3} m )
D. ( 30 sqrt{3} m )
10
28A man from top of a 100 meters high towers sees a car moving towards the tower at an angle of depression of ( 30^{0} ) After some time, the angle of
depression becomes ( 60^{0} . ) The distance (in meters) travelled by the car during this time is
A. ( 100 sqrt{3} )
3
в. ( frac{200 sqrt{3}}{3} )
c. ( frac{100 sqrt{3}}{3} )
D. ( 200 sqrt{3} )
10
29(U) uyu
11.
AB is a vertical pole with B at the
ZU V
ical pole with B at the ground level and A at the
top. A man finds that the angle of elevation of the po
a certain point on the ground is 60°. He moves away
” the pole along the line BC to a point D such that CDE
from D the angle of elevation of the point A is 45°. Then
the height of the pole is
[2008]
(a)
7/3 1
2 13-1″
-m
(13+1)
10
30The angle of elevation of the top of a
tower from a point on the ground ( 30 mathrm{m} )
away fro the foot of the tower is ( 30^{circ} . ) The
height of the tower is
tower
( mathbf{A} cdot 30 mathrm{m} )
B. ( 10 sqrt{3} mathrm{m} )
( mathbf{c} cdot 10 mathrm{m} )
D. ( 10 sqrt{2} mathrm{m} )
10
31The shadow of the tower standing on a
level ground is found to be 60 meters
longer when the suns altitude is ( 30^{circ} )
than when it is ( 45^{circ} . ) The height of the
tower is
( A cdot 60 m )
в. ( 30 m )
( c cdot 60 sqrt{3} m )
D. ( 20(sqrt{3}+1) m )
10
32Find the length of the shadow on the
ground of a pole of height ( 6 m ) when
the angle of elevation ( theta ) of the Sun is such that ( tan theta=frac{3}{4} )
10
339 IULIE
A bird is sitting on the top of a vertical pole 20 m high and its
elevation from a point on the ground is 45°. It flies off
horizontally straight away from the point O. After one second
the elevation of the bird from O is reduced to 30°. Then the
speed (in m/s) of the bird is
JJEEM 2014
(a) 2012
(b) 20(3-1)
(c) 40 (√2-1)
(d) 40/13-√2)
10
34e centre of a circular park. A and B are
ndary of the park such that AB (a)
(a) U P. (b) A P. (c) A.
A tower stands at the centre of a circular park.
two points on the boundary of the park such tha
subtends an angle of 60° at the foot of the tower, an
angle of elevation of the top of the tower from A or B
The height of the tower is
[2007]
(a) a/v3 (b) a 3 (C) 2a/13 (d) 2av3.
le top of the tower from A or B is 30°.
10
35The horizontal distance between two
trees of different height is 60 m. the angle of depression of the top of the first
tree when seen from the top of the
second tree is ( 45^{0} . ) If the height of the
second tree is ( 80 mathrm{m} ), find the length of
the tree
10
3665. The angle of elevation of the top
of a vertical tower situated per-
pendicularly on a plane is ob-
served as 60° from a point P on
the same plane. From another
point 9, 10m vertically above the
point P, the angle of depression
of the foot of the tower is 30°.
The height of the tower is
(1) 15 m (2) 30 m
(3) 20 m (4) 25 m
10
37TJU (b) W
( 120
A person standing on the bank of
standing on the bank of a river observes that the
angle of elevation of the top of a tree o
of elevation of the top of a tree on the opposite bank
of the river is 60° and when here
ver is 60° and when he retires 40 meters away from
ne tree the angle of elevation becomes 30°. The breadth of
the river is
[2004]
(a) 60 m (b) 30 m (c) 40 m (d) 20 m
10
38Illustration 3.37 A ladder rests against a wall making an
angle a with the horizontal. The foot of the ladder is pulled
away from the wall through a distance x, so that it slides
a distance y down the wall making an angle B with the
horizontal. Prove that x = y tan –
a+ß
10
3915 V, Find
and we litigio
Wer AB leans towards west making an angle a with the
vertical. The angular elevation of B, the topmos
Is ß as observed from a point C due west of A at a
distance d from A. If the angular elevation of B from a point D
due east of Cat a distance 2d from C is y, then prove that 2 tan
a=-cot ß + coty.
(1994 – 4 Marks)
10
4075. From two points on the ground
and lying on a straight line
through the foot of a pillar, the
two angles of elevation of the top
of the pillar are complementary
to each other. If the distances of
the two points from the foot of
the pillar are 12 metres and 27
metres and the two points lie on
the same side of the pillar, then
the height (in metres) of the pil-
lar is
(1) 12 (2) 18
(3) 15
(4) 16
10
41A bird flies in a circle on a horizontal
plane. An observer stands at a point on the ground. Suppose ( 60^{circ} ) and ( 30^{circ} ) are the maximum and the minimum angles
of elevation of the bird and that they
occur when the bird is at the points ( mathrm{P} ) and ( Q ) respectively on its path. Let ( theta ) be the angle of elevation of the bird when it is a point on the are of the circle exactly midway between ( P ) and ( Q ). Find the numerical value of ( tan ^{2} theta ) (Assume that
the observer is not inside the vertical
projection of the path of the bird).
10
42The angle of elevation of the top of a
tower from a point ( 20 m ) away from its
base is ( 60^{circ} . ) The height of the tower is-
( A cdot 2 sqrt{3} m )
В. ( -20 sqrt{3} mathrm{m} )
c. ( sqrt{3} m )
D. ( 20 sqrt{3} ) m
10
43A kite is flying with the string inclined
at ( 45^{circ} ) to the horizontal If the string is
straight and ( 50 mathrm{m} ) long the height at which the kite is flying is
A ( .25 sqrt{2} m )
в. ( 50 sqrt{2} mathrm{m} )
c. ( 25 mathrm{m} )
D. 50 ( m )
10
44The angle of elevation of the top of a vertical tower from a point ( A, ) due east
of it is ( 45^{circ} . ) The angle of elevation of the
top of the same tower from a point ( boldsymbol{B} )
due south of ( A ) is ( 30^{circ} . ) If the distance
between ( A ) and ( B ) is ( 54 sqrt{2} m, ) then
the height of the tower (in metres), is :
A. ( 36 sqrt{3} )
в. 108
( c .54 )
D. ( 54 sqrt{3} )
10
45A pole has to be erected at a point on the boundary of a circular park of
diameter 13 meters in such a way that the difference of its distance from two
diametrically opposite fixed gates ( boldsymbol{A} ) and ( B ) on the boundary is 7 meters. It is possible to do so? If yes, at what distances from the two gates should the
pole be erected?
A. ( 6.71 m )
в. ( 0.71 m )
c. ( 1.51 m )
D. ( 5.51 mathrm{m} )
10
46Illustration 3.17
The upper – th portion of a vertical pole
subtends an angle o such that tan 0 = – at a point in the
horizontal plane through its foot and at a distance 40 m from
the foot. Find the possible height of the vertical pole.
10
47A kite is flying with the string inclined
at ( 75^{circ} ) to the horizon. If the length of the
string is ( 25 m ), then height of the kite is :
A ( cdotleft(frac{25}{2}right)(sqrt{3}-1)^{2} )
B. ( left(frac{25}{2}right)left(frac{sqrt{3}+1}{sqrt{2}}right) )
c. ( left(frac{25}{2}right)(sqrt{3}+1)^{2} )
D. ( left(frac{25}{2}right)(sqrt{6}+sqrt{2}) )
10
48The angle of elevation of a jet plane from a point ( A ) on the ground is ( 60^{0} ). After a
flight of 15 seconds, the angle of elevation changes to ( 30^{0} . ) If the plane at a constant height of ( 1500 sqrt{3} m ), then the
speed of jet plane is :
A. ( 200 mathrm{m} / mathrm{s} )
B. 1980 m/s
c. ( 240 mathrm{m} / mathrm{s} )
D. 220 ( mathrm{m} / mathrm{s} )
10
49Fill in the blanks
A tower is observed from two stations ( A )
and ( B ) where ( B ) is East of ( A ) at a distance
100 metres. The tower is due North of ( A )
and due North-West of
B. The angles of elevation of the tower from ( A ) and ( B ) are
complementary, the height of the tower
is
10
50A vertical tower OP stands at the centre
O of a square ABCD. Let h and b denote the length OP and AB respectively.
Suppose ( angle A P B=60^{circ}, ) then the
relationship between h and b can be
expressed as:
( mathbf{A} cdot 2 b^{2}=h^{2} )
B ( cdot 2 h^{2}=b^{2} )
( mathbf{c} cdot 3 b^{2}=2 h^{2} )
D. ( 3 h^{2}=2 b^{2} )
10
51QR is a triangular park with PQ=PR=200 m. AT.V. tov
stands at the mid-point of QR. If the angles of elevation of
the top of the tower at P, Q and R are respectively 45°, 300
and 30°, then the height of the tower (in m) is :
[JEEM 2018
(a) 50
(b) 100/3
(d) 100
502
10
52The angle of elevation of the top of a building from the foot of the tower is 30
and the angle of elevation of the top of
the tower from the foot of the building is
( 60^{circ} . ) If the tower is 48 meters high, find
the height of the building (in ( mathrm{m} ).)
10
5317.
dls leeu. THU MU UUU
ABC is a triangular park with AB=AC = 100 m. A television
tower stands at the midpoint of BC. The angles of elevation
of the top of the tower at A, B, C are 45°, 60°, 60°, respectively.
Find the height of the tower.
(1989 – 5 Marks)
Doints and on
10
54V )
14. If the angles of elevation of the top of a tower from three
collinear points A, B and C, on a line leading to the foot of the
tower, are 30°, 45° and 60° respectively, then the ratio, AB
BC, is:
(JEE M 2015)
(a) 1:13
(b) 2:3
(c) 13:1
(d)
13:12
10
5565. The shadow of a tower standing
on a level ground is found to be
40 metre longer when the suns
altitude is 30° than when it is
60°. Find the length of the tower.
(1) 20/3 m (2) 10 m
(3) 10/3 m
(4) 20 m
10
56From the top of a vertical tower, the
angles of depression of two cars, in the same straight line with the base of tower, at an instant are found to be 45 and ( 60 . ) If the cars are ( 100 mathrm{m} ) apart and
are on the same side of the tower, find the height of the tower. [ Use ( sqrt{3}=1.73] )
10
57P. Let a vertical tower AB have its end A on the level ground
Let C be the mid-point of AB and P be a point on the ground
such that AP=2AB. If ZBPC=B, then tan ß is equal to :
[JEE M 2017
10
58As observed from the top of a ( 60 mathrm{m} ) high lighthouse from the sea-level, the angles of depression of two ships are
( 30^{circ} ) and ( 45^{circ} . ) If one ship is exactly behind the other on the same side of the
lighthouse. Find the distance between
the two ships.
A. ( 20 m )
в. ( 46.5 mathrm{m} )
c. ( 54.9 m )
D. ( 60.1 mathrm{m} )
10
59V.
VE
A vertical pole stands at a point on a horizontal ground. A
and B are points on the ground, d meters apart. The pole
subtends angles a and B at A and B respectively. AB
subtends an angle y at Q. Find the height of the pole.
(1982 – 3 Marks)
10
6067. An observer, 1.5 metre tall,
standing on a 5 metre height
house looks a 28.5 metre distant
pole of height 35 metre. What will
be the angle of elevation ?
Blo
(4) 6
10
6103
71. A man stands at a point A on the
bank of a river and looks at the
top of a tree which is exactly op-
posite to him on the other bank.
The angle of elevation is 45º. He
then walks 200 m at right angles
to the bank and away from it to
the point B. From B he looks at
the top of the tree and the angle
of elevation as 30°. The height of
the tree is:
(1) 10(73 +1) m
(2) 100(13 – 1)m
(3) 88 (13 + 1)m
(4) 100 (73 +1) m
10
6270.
Two poles of equal heights are
standing opposite to each other
on either side of a road which is
100m wide. From a point be-
tween them on road, angles of
elevation of their tops are 30° and
60°. The height of each pole in
metre, is
(1) 25/3 (2) 20.3
(3) 28/3
(4) 30/3
10
63The angles of elevation of the top of a rock from the top and foot of a ( 100 mathrm{m} )
high tower are respectively ( 30^{circ} ) and ( 45^{circ} ) Find the height of the rock.
10
64A tree breaks due to storm and the
broken part bends so that the top of the
tree touches the ground by making ( 30^{circ} )
angle with the ground. The distance between the foot of the tree and the top
of the tree on the ground is 6 m. Find the height of the tree before falling down.
10
6582. The distance between the two parallel lines is 1 unit. A
point ‘A’ is chosen to lie between the lines at a distance
d’ from one of them. Triangle ABC is equilateral with B
on one line and C on the other parallel line. The length of
the side of the equilateral triangle is
d² – d+1
b. 21
3
a. Vd2 + d +1
c. 2d² – d+1
d. ſd² – d+1 ne
10
6673. The shadow of a tower standing
on a level plane is found to be
40m longer when the sun’s alti-
tude is 45°, than when it is 60°
The height of the tower is
(1) 30 (3+3 metre
(2) 40 (3+13) metre
(3) 20 (3+ /3) metre
(4) 10 (3+13) metre
10
67. The two banks of a canal are
straight and parallel. A, B, C are
three persons of whom A stands
on one bank and B and C on the
opposite banks. B finds the an-
gle ABC is 30°, while C finds that
the angle ACB 60°. If B and C
are 100 metres apart, the