7 min readConcept sheetPart II (2 marks); Part III (5 marks)

Heights and distances: angle of depression

Looking down from a height, the angle between the horizontal and your line of sight is the angle of depression. It equals the angle of elevation from below.

Angle of depressionThe angle between the horizontal line at the eye and the line of sight, when the object is below the eye

Key factAngle of depression from A to B = angle of elevation from B to A

ReasonThey are alternate angles between parallel horizontal lines

Why depression equals elevation

Suppose you stand at the top A of a cliff and look down at a boat C. The horizontal line through your eye is parallel to the ground. The line of sight AC cuts both of these lines. So the angle of depression at A and the angle at C inside the triangle are alternate angles, and alternate angles are equal. This lets you put the angle inside the right triangle, at the object on the ground.

How to draw the figure

1
Draw a horizontal line for the ground and a vertical line for the cliff or tower. Mark the top A and the foot B.
2
At A, draw a dotted horizontal line, parallel to the ground.
3
Mark the object C on the ground and join A to C (the line of sight).
4
Mark the angle of depression between the dotted line and AC. It is not the angle between AC and the cliff.
5
Mark the same angle at C, inside the triangle (alternate angles).
6
Mark the right angle at B. Label the height AB and the distance BC.

Elevation or depression?

Angle of elevationThe observer looks up at an object above the eye. The angle is measured up from the horizontal.

Angle of depressionThe observer looks down at an object below the eye. The angle is measured down from the horizontal at the eye.

Both angles are measured from the horizontal line, never from the vertical.

Example 1: a boat seen from a cliff

Problem

From the top of a cliff 60 m high, the angle of depression of a boat is 30°. How far is the boat from the foot of the cliff? (√3 = 1.732)

Working

Formula: tan θ = opposite side/adjacent side
Let AB = 60 m be the cliff and C the boat. The angle of depression is 30°, so ∠ACB = 30° (alternate angles).
tan 30° = AB/BC = 60/BC
1/√3 = 60/BC
BC = 60√3 = 60 × 1.732 = 103.92

Answer

The boat is 103.92 m from the foot of the cliff.

Example 2: a car seen from a balcony

Problem

From a balcony 24 m above the ground, a boy sees a car parked on the road at an angle of depression of 60°. How far is the car from the building? (√3 = 1.732)

Working

Formula: tan θ = opposite side/adjacent side
Let AB = 24 m (the balcony height) and C the car. ∠ACB = 60° (alternate angles).
tan 60° = AB/BC = 24/BC
√3 = 24/BC, so BC = 24/√3
BC = 24√3/3 = 8√3 = 8 × 1.732 = 13.856 m

Answer

The car is about 13.86 m from the building (rounded to two decimal places).

In real life

A lighthouse keeper on the Tamil Nadu coast, or a lifeguard on a tall chair at Marina Beach, can judge how far away a boat is from one angle of depression and the known height.

Exam angle

2 marks: “From the top of a cliff 50 m high, the angle of depression of a boat is 45°. How far is the boat from the foot of the cliff?” tan 45° = 50/d, so 1 = 50/d and **d = 50 m**.

2 marks: “Why is the angle of depression from A to B equal to the angle of elevation from B to A?” The horizontal line at A is parallel to the ground at B. The line of sight AB cuts both, so the two angles are **alternate angles**, which are equal.

5 marks: “From the top of a hill 40√3 m high, the angle of depression of a bus stop on the road below is 30°. Find the distance of the bus stop from the foot of the hill.” The angle at the bus stop is 30° (alternate angles). tan 30° = 40√3/d, so d = 40√3 × √3 = **120 m**.

Remember it

Depression: look down, then drop the angle to the ground (alternate angles).