A cliff on the bank of a river 87 m high. A boat on the river is 22 m away from the cliff. Calculate, correct…

Analyze the question: The cliff is 87 m high and the boat is 22 m horizontally from the base of the cliff. We need the angle of depression from the top of the cliff to the boat.

Concept: The angle of depression is measured downward from the horizontal at the observer's eye level. Because that horizontal line is parallel to the river surface, the angle of depression equals the angle of elevation of the cliff top as seen from the boat (alternate angles).

Solution:

Let θ be the angle of depression. In the right-angled triangle:

• Opposite side (height of the cliff) = 87 m
• Adjacent side (distance of the boat from the cliff) = 22 m

tan θ = opposite / adjacent = 87 / 22 = 3.9545

θ = tan⁻¹(3.9545) ≈ 75.8°, which is 76° to the nearest degree.

Check: tan 76° ≈ 4.01 and tan 75° ≈ 3.73, so 3.95 lies between them, closer to 76°. ✔

Answer: The angle of depression is 76°, which is option A.

Tip: In depression and elevation questions, draw the right-angled triangle first. If the height is much bigger than the horizontal distance, the angle must be steep (above 45°). That alone removes options C and D here. Also remember that the angle of depression is measured from the horizontal, not from the cliff face. Measuring it from the cliff face would give the wrong answer, 14°.