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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
Solution
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`
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