Question

A tower subtends an angle α at a point A in the plane of its base and the angle of 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

h = b tan α cot β (Proved)

Step-by-step explanation:

See the attached diagram.

A is the point of observation of the top of the tower BD and point C is b meters above point A.

Taking the right triangle Δ ABC, $\tan \beta = \frac{AC}{AB} = \frac{b}{AB}$

⇒ AB = b cot β ........... (1)

Again, taking the right triangle Δ ABD,

$\tan \alpha = \frac{BD}{AB} = \frac{h}{b\cot \beta}$ {From equation (1)}

⇒ h = b tan α cot β (Proved)