# Some Applications of Trigonometry

**Exercise 1**

**1.** A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30^{0} (see figure).

**2.** A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30^{0} with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

**3.** A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m and is inclined at an angle of 30^{0} to the ground, whereas for elder children, she wants to have a steep slide at a height of 3 m and inclined at an angle of 60^{0} to the ground. What should be the length of the slide in each case?

**4.** The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the lower is 30^{0}. Find the height of the tower.

**5.** A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60^{0}. Find the length of the string, assuming that there is no slack in the string.

**6.** A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30^{0} to 60^{0} as he walks towards the Find the distance he walked towards the building.

**7.** From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45^{0} and 60^{0} respectively. Find the height of the tower.

**8.** A statue, 1.6 m tall, stands on the top of a postal. From a point on the ground, the angle of elevation of the top of the statue is 60^{0} and from the same point the angle of elevation of the top of the pedestal is 45^{0}. Find the height of the pedestal.

**9.** The angle of elevation of the top of a building from the foot of the tower is 30^{0} and the angle of elevation of the top of the tower from the foot of the building is 60^{0}. If the tower is 50 m high, find the height of the building.

**10. ** Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60^{0} and 30^{0} respectively. Find the height of the poles and the distances of the point from the poles.

**11. ** A TV tower stands 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^{0}. 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^{0} (see figure). Find the height of the tower and the width of the canal.

**12.** From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60^{0} and the angle of depression of its foot is 45^{0}. Determine the height of the tower.

**13.** As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30^{0} and 45^{0}. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between two ships.

**14.** 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 any distant is 60^{0}. After some time, the angle of elevation reduces to 30^{0} (see figure). Find the distance travelled by the balloon during the interval.

**15. ** A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30^{0}, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60^{0}. Find the time taken by the car to reach the foot of the tower from this point.

**16. ** The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is 6 m.