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 No | Questions | Class |
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1 | A 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 |

3 | Two 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 |

4 | 51. 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 |

5 | Illustration 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 |

6 | 66. 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 |

7 | A 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 |

8 | The 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 |

9 | A 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 |

10 | True 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 |

11 | A 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 |

12 | Illustration 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 |

13 | A 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 |

14 | 67. 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 |

15 | Find the angle of evelation of the sun when the shadow of a pole ( h ) meter high is ( sqrt{3} h ) meters long | 10 |

16 | 61. 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 |

17 | 74. 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 |

18 | A 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 |

19 | 75. 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 |

20 | 87. 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 |

22 | 68. 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 |

23 | 52. 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 |

24 | 75. 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 |

25 | A 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 |

26 | 27 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 |

27 | Choose 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 |

28 | A 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) © 748 (13–19m () 754 | 10 |

30 | The 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 |

31 | The 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 |

32 | Find 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 |

33 | 9 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 |

34 | e 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 |

35 | The 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 |

36 | 65. 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 |

37 | TJU (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 |

38 | Illustration 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 |

39 | 15 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 |

40 | 75. 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 |

41 | A 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 |

42 | The 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 |

43 | A 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 |

44 | The 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 |

45 | A 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 |

46 | Illustration 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 |

47 | A 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 |

48 | The 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 |

49 | Fill 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 |

50 | A 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 |

51 | QR 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 |

52 | The 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 |

53 | 17. 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 |

54 | V ) 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 |

55 | 65. 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 |

56 | From 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 |

57 | P. 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 |

58 | As 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 |

59 | V. 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 |

60 | 67. 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 |

61 | 03 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 |

62 | 70. 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 |

63 | The 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 |

64 | A 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 |

65 | 82. 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 |

66 | 73. 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 breadth of the canal is (1) 25 T2 metres (2) 20.3 metres (3) 25 3 metres 20 (4) T5 metres | 10 |

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