From the top of a vertical cliff 20m high, a boat at sea can be sighted 75m away and on the same horizontal…

Analyze the question

A cliff is 20 m high. A boat is 75 m away. The question says "75m away and on the same horizontal position as the cliff". The usual reading is that the boat is 75 m horizontally from the foot of the cliff. We need the angle of depression from the top of the cliff to the boat.

Concept

The angle of depression from the top of the cliff equals the angle of elevation of the top of the cliff from the boat (alternate angles). In the right-angled triangle:

• opposite side = height of cliff = 20 m
• adjacent side = horizontal distance = 75 m

So tan θ = opposite/adjacent.

Solution (horizontal distance = 75 m)

tan θ = 20/75 = 0.2667

θ = tan⁻¹(0.2667) ≈ 14.93° ≈ 15°

Check the alternative reading

If 75 m were the line-of-sight distance, then sin θ = 20/75 = 0.2667, so θ ≈ 15.46° ≈ 15°. This also gives 15°, so both readings agree to the nearest degree.

Option C (16°) would only come from a miscalculation, and 75° is the angle at the top of the cliff measured from the vertical (90° − 15°), a common trap.

Answer: The angle of depression is about 15°, which is option D.

Exam tip: Always draw the diagram. The angle of depression is measured from the horizontal line at the observer's eye level, not from the vertical. When the two readings of a question give the same answer, you can be confident. Also remember that "SOH CAH TOA" picks the correct ratio: here you have opposite and adjacent, so use tan.