The angle of elevation of a cliff from a fixed point A is θ. After going up adistance of k metres towards the top of the cliff at an angle φ, itis found that the angle of elevation is α. Show that the height ofthe cliff in metre is .

Question

Philoid · Accepted Answer

Given:

Angle of elevation from fixed point A = θ

Distance covered = k m

Angle after the distance k metres is covered = ϕ

In right angled Δ FED,

FE = k m

sin φ = ED / FE

∴ ED = FE sin φ

= k sin φ

Again, cos φ = FD / FE

∴ FD = FE cos φ

= k cos f .....................(1)

In right angled Δ AFC,

FC / AC = cot θ

FC = AC cot q................................... (2)

FC - FD = DC

But, DC = EB

∴ EB = FC – FD

Thus, EB = AC cot θ - k cos f …………………. (from (1) and (2))

Now, AB = AC - BC

But, BC = ED = k sin φ

∴ AB = AC - k sin φ

In right angled Δ AEB, cot α = EB / AB

AC cot θ - k cos φ = AC cot α - k sin φ cot α

AC cot θ - AC cosα = k cos φ - k sin φ cot α

AC (cot θ - cot α ) = k cos φ - k sin φ cotα

Hence proved the height of the cliff in metres is