Some Applications of Trigonometry

• Line of sight : When an observer looks from a point O at an object P, then the line OP is called the line of sight.
• Angle of elevation : The angle which the line of sight makes with the horizontal line through O is called the angle of elevation of P, as seen from O; i.e., ∠XOP.
Or the angle of elevation of an object viewed, is the angle formed by the line of sight with the horizontal when it is above the horizontal level. i.e. the case when we raise our head to look the object.

• Angle of depression : The angle which the line of sight makes with the horizontal line through O is called the angle of depression of P, as seen from O.
Or the angle of depression of an object viewed, is the angle formed by the line of sight with the horizontal when it is below the horizontal level. i.e., the case when we lower our head to look at the objec

• Altitude of the sun : The altitude of the sun is simply the angle of elevation of the sun.

Important Questions

1. A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, fnd the height of the wall.
2. The shadow of a tower is 30 m long, when the sun’s elevation is 30°. What is the length of the shadow, when sun’s elevation is 60° ?
3. The angle of elevation of the top of a tower from two points distant a and b from the base and in the same straight line with it are complementary. Prove that the height of tower is $\inline \dpi{120} \sqrt{ab}$.
4. A circus artist is climbing a rope 12 m long which is tightly stretched and tied from the top of a vertical pole to the ground. fnd the height of the pole if the angle made by the rope with the ground is 30°.
5. Two poles of equal heights are standing opposite to each other on either side of a road, which is 100 metres wide. From a point between them on the road, the angles of elevation of their tops are 30° and 60°. Find the position of the point and also, the heights of the poles.
6. A tree breaks due to the storm and the broken part bends so that the top of the tree touches ground making an angle of 30° with the ground. The distance from the foot of the tree to the point where the top touches the ground, is 10 metres. Find the height of the tree.
7. A boy is standing on the ground and ﬂying a kite with 100 m of string at an elevation of 30°. Another boy is standing on the roof of a 20 m high building and is ﬂying his kite at an elevation of 45°. Both the boys are on the opposite sides of both the kites. Find the length of the string that the second boy must have so that the two kites meet.
8. At the foot of a mountain the elevation of its summit is 45°. After ascending 1000 m towards the mountain up a slope of 30° inclination, the elevation is found to be 60°. Find the height of the mountain.
9. The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60°. At a point Y, 40 m vertically above X, the angle of elevation is 45°. Find the height of the tower PQ and the distance XQ.
10. A round balloon of radius r subtends an angle α at the eye of the observer while the angle of elevation of its centre is β. Prove that the height of the centre of the balloon is r sin β cosec α/2.
11. The angle of elevation of a cliff from a fxed point is θ. After going up a distance of k metres towards the top of the cliff at an angle of φ, it is found that the angle of elevation is α. Show that the height of the cliff is $\inline \dpi{120} \frac { k ( \cos \phi - \sin \phi \cot \alpha ) } { \cot \theta - \cot \alpha }$ metres.
12. If the angle of elevation of a cloud from a point h metres above a lake is α and the angle of depression of its reﬂection in the lake from the same point is β, prove that the height of the cloud is $\inline \dpi{120} \frac { h ( \tan \beta + \tan \alpha ) } { ( \tan \beta - \tan \alpha ) }$.
13. A man on a cliff observes a boat at an angle of depression of 30° which is approaching the shore to the point immediately beneath the observer with a uniform speed. Six minutes later, the angle of depression of the boat is found to be 60°. Find the time taken by the boat to reach the shore.
14. A ladder rests against a wall at an angle α to the horizontal. Its foot is pulled away from the wall through a distance a, so that it slides a distance b down the wall making an angle β with the horizontal. Show that $\inline \dpi{120} \frac { a } { b } = \frac { \cos \alpha - \cos \beta } { \sin \beta - \sin \alpha }$.
15. The angles of elevation of the top of a hill at the city centres of two towns on either side of the hill are observed to be 30° and 60°. If the distance up-hill from the frst city centre is 9 km, find in kilometres the distance up-hill from the other city centre.
16. An aeroplane when 3000 m high passes vertically above another aeroplane at an instance when their angles of elevation at the same observation point are 60° and 45° respectively. How many metres higher is the one than the other ?
17. The distance between two vertical poles is 60 m. The height of one of the poles is double the height of the other. The angles of elevation of the tops of the poles from the middle point of the line segment joining their feet are complementary to each other. Find the heights of the poles.
18. The angles of elevation of a cloud from a point h metres above a lake is 30° and the angle of depression of its reﬂection in the lake is 45°. If the height of the cloud be 200 m, find h.
19. A bird is perched on the top of a tree 20 m high and its angle of elevation from a point on the ground 45°. The bird ﬂies off horizontally straight away from the observer and in one second the angle of elevation of the bird reduces to 30°. Find the speed of the bird.
20. From the top of a tower, the angles of depression of two objects on the same side of the tower are found to be α and β (α > β). If the distance between the objects is a metres, show that the height of the tower is $\inline \dpi{120} \frac { a \tan \alpha \tan \beta } { \tan \alpha - \tan \beta }$.
21. A vertical tower stands on a horizontal plane and is surmounted by a vertical ﬂagstaff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the ﬂagstaff are α and β respectively. Prove that the height of the tower is  $\inline \dpi{120} \frac { h \tan \alpha } { \tan \beta - \tan \alpha }$.
22. A tree standing on a horizontal plane is leaning towards east. At two points situated at distance a and b exactly due west of it, the angles of elevation of the top are α and β respectively. Prove that the height of the top of the tree from the ground is $\inline \dpi{120} \frac { ( b - a ) \tan \alpha \tan \beta } { \tan \alpha - \tan \beta }$.
23. At a point on a level plane, a tower subtends an angle α and a man a metres tall standing on its top subtends an angle β. Prove that the height of the tower is $\inline \dpi{120} \frac { a \sin \alpha \cos ( \alpha + \beta ) } { \sin \beta }$.
24. A tower stand vertically on a bank of a canal from a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is 60°, from another point 20 m away from this point on the line joining this point to the foot of the tower, the angle of elevation of the top of the tower is 30°. Find the height of the tower and the width of the canal.
25. The angle of elevation of the top of a building from the foot of a tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high. Find the height of the building.
26. A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at that instant is 60°. After some time the angle of elevation reduces to 30°. Find the distance travelled by the balloon during the interval.
27. From a window, 60 m high above the ground, of a house in a street, the angles of elevation and depression of the top and foot of another house on the opposite side of the street are 60° and 45° respectively. Show that the height of the opposite house is 60 (1+ √3 ) metres.
28. On a horizontal plane there is a vertical tower with a ﬂag pole on the top of the tower. At a point 9 metres away from the foot of the tower the angle of elevation of the top and bottom of the ﬂag pole are 60° and 30° respectively. Find the height of the tower and ﬂag pole mounted on it.
29. A boy standing on a horizontal plane fnds a bird ﬂying at a distnce of 100 m from him at an elevation of 30°. A girl standing on the roof of 20 metre high building, fnds the angle of elevation of the same bird to be 45°. Both the boy and the girl are on opposite sides of the bird. Find the distance of bird from the girl.
30. Two men on either side of a cliff, 60 m high, observe the angles of elevation of the top of the cliff to be 45° and 60° respectively. Find the distance between two men.
31. A man standing on the deck of the ship which is 10 m above the sea level, observes the angle of elevation of the top of the cloud as 30° and angle of depression of its reﬂection in the sea was found to be 60°. Find the height of the cloud and also the distance of the cloud from the ship.
32. A person standing on the bank of a river observes that the angle of elevation of the top of the tree standing on the opposite bank is 60°. When he moves 30 m away from the bank, he finds the angle of elevation to be 30°. Find the height of the tree and the width of the river.
33. The angle of elevation θ of the top of a light-house, as seen by a person on the ground, is such that tan θ = 5/12. When the person moves a distance of 240 m towards the light-house, the angle of elevation becomes φ such that tan φ = 3/4. Find the height of the light-house.
34. A man on the top of a vertical tower observes a car moving towards the tower. If it takes 12 minutes for the angle of depression to change from 30° to 45°, how soon after this the car will reach the tower?