# Class 10 – Some Applications of Trigonometry – Important Questions

Some Applications of Trigonometry – Important Questions

1. A person, standing on the bank of a river, observes that the angle subtended by a tree on the opposite bank is 60°. When he retreated 20 m from the bank, he finds the angle to be 30°. Find the height of the tree and the breadth of the river.
2. A tree is broken by the wind. The top struck the ground at an angle of 30° and at a distance of 30 meters from the roots. Find the whole height of the tree.
3. The angles of elevation of the top of a tower from two points at a distance p and q meters from the base and in the same straight line with it are complementary. Prove that the height of the tower is √pq .
4. At a point on level ground, the angle of elevation of a vertical tower is found to be such that its tangent is 5/12 .On walking 192 meters towards the tower, the tangent of the angle of elevation is 3/4. Find the height of the tower.
5. A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite side of the bank is 60°. When he moves 40 meters 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 .
6. A man on the top of a vertical tower observes a car moving at a uniform speed coming directly towards it. If it takes 12 minutes for the angle of depression to change from 30° to 45°, how soon after this, will the car reach the tower? Give your answer to the nearest second .
7. The shadow of a flag – staff is three times as long as the shadow of the flag – staff when the sun rays meet the ground at an angle of 60°. Find the angle between the sun rays and the ground at the time of longer shadow.
8. An aeroplane at an altitude of 200 m observes the angles of depression of opposite points on the two banks of river to be 45° and 60°. Find the width of the river.
9. Two pillars of equal height and on either side of a road, which is 100 m wide . The angles of elevation of the top of the pillars are 60° and 30° at a point on the road between the pillars. Find the position of the point between the pillars and the height of each pillar.
10. The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 30 seconds, the angle of elevation changes to 30°. If the jet plane is flying at a constant height of 3600√3 m, find the speed of the jet plane.
11. If the angle of elevation of a cloud from a point h meters above a lake is α and the angle of depression of its reflection in the lake is β. Prove that the height of the cloud is h(tan β + tan α)/(tan β – tan α).
12. 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.
13. The angle of elevation of a cliff from a fixed point is θ. After going up a distance of K meters 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 k(cos φ – sin φ cot α)/(cot φ – cot α) meter.