Question

A tower subtends an angle 𝛼 at a point A in the plane of its base and the angle if depression of the foot of the tower at a point b metres just above A is β. Prove that the height of the tower is b tan α cot β

Answer 1

Let h be the height of tower CD. The tower CD subtends an angle α at a point A. And the angle of depression of foot of tower at a point b meter just above A is β.

Let AC = x and ∠ACB = β, ∠CAD = α

Here we have to prove height of the tower is b tan α cot β

We have the corresponding figure as follows

So we use trigonometric ratios.

In ΔABC

`=> tan beta = (AB)/(AC)`

`=> tan beta = b/x`

`=> x = b/(tan beta)`

`=> x = b cot beta`

Again in ΔACD

`=> tan σ = (CD)/(AC)`

`=> tan alpha = h/x

`=> h = xtan alpha`

`=> h = b tan alpha cot beta`

Hence the height of tower is `b tan alpha cot beta`