Some Applications of Trigonometry Questions

 Some Applications of Trigonometry

Q. A spherical balloon of radius `r` subtends an angle `theta` at the eye of an observer. If the angle of elevation of its centre is `phi`, find the height of the centre of the balloon.

Q. From a balloon vertically above a straight road, the angles of depression of two cars at an instant are found to be 45° and 60°. If the cars are 100 m apart, find the height of the balloon.

Q. The angle of elevation of a cloud from a point `h` metres above the surface of a lake is `theta` and the angle of depression of its reflection in the lake is `phi`. Prove that the height of the cloud above the lake is

`h(frac{tan phi + tan theta}{tan phi - tan theta})`

Q. The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60° and the angle of elevation of the top of the second tower from the foot of the first tower is 30°. Find the distance between the two towers and also the height of the other tower.

Q. From the top of a tower `h` m high, the angles of depression of two objects, which are in line with the foot of the tower are `alpha` and `beta (beta  > alpha)`. Find the distance between the two objects.

Q. A window of a house is `h` metres above the ground. From the window, the angles of elevation and depression of the top and the bottom of another house situated on the opposite side of the lane are found to be `alpha` and `beta`, respectively. Prove that the height of the other house is `h ( 1 + tan alpha cot beta )` metres.

Case Studies 

1. Trigonometry in the form of triangulation forms the basis of navigation, whether it is by land, sea or air. GPS a radio navigation system helps to locate our position on earth with the help of satellites.

A guard, stationed at the top of a 240m tower, observed an unidentified boat coming towards it. A clinometer or inclinometer is an instrument used for measuring angles or slopes(tilt). The guard used the clinometer to measure the angle of depression of the boat coming towards the lighthouse and found it to be 30°.

(Lighthouse of Mumbai Harbour. Picture credits - Times of India Travel)

i) Make a labelled figure on the basis of the given information and calculate the distance of the boat from the foot of the observation tower.

ii) After 10 minutes, the guard observed that the boat was approaching the tower and its distance from tower is reduced by `240(sqrt3 - 1)` m. He immediately raised the alarm. What was the new angle of depression of the boat from the top of the observation tower?