Bearings – SS2 Mathematics Lesson Note (Third Term)

Angles of elevation and depression are angles measured from the eye level of an observer. The angle of elevation is measured when looking up at an object, and the angle of depression when looking down at an object. Problems on them are solved with the basic trigonometric ratios, and sometimes with the sine rule and cosine rule.

Behavioural objectives

By the end of the lesson, students should be able to:

• define angle of elevation and angle of depression;
• state the sine rule and the cosine rule;
• solve problems on angles of elevation and depression using trigonometric ratios;
• give answers to a stated degree of accuracy.

What is an angle of elevation?

This is the angle formed between the normal eye level and the line through which the observer views an object above.

In triangle ABC, angle ACB = θ = angle of elevation.

What is an angle of depression?

This is the angle formed between the eye level of the observer and the line of sight to an object below, when the observer is above the object in view.

Angle ABC = θ = angle of depression.

How are these angles solved?

The angle of elevation and the angle of depression are alternate angles, so they are equal. Problems involving them can be solved with the basic trigonometric ratios. In some cases the sine rule and the cosine rule can be applied.

Sine rule

For triangle ABC:

a / Sin A = b / Sin B = c / Sin C

Cosine rule

1. a² = b² + c² – 2bc Cos A
2. b² = a² + c² – 2ac Cos B
3. c² = a² + b² – 2ab Cos C

Worked examples

Example 1

A ladder 50 m long rests against a vertical wall. If the ladder makes an angle of 65° with the ground, find the distance between the foot of the ladder and the wall.

Solution

Ladder = QR, wall = QP.

Distance between the foot of the ladder and the wall = PR.

Cos 65° = PR / 50

Cross multiplying:

PR = 50 × Cos 65° = 50 × 0.4226

PR = 21.13 m

Example 2

The angle of depression of an object on the ground from the top of a tower 60 m high is 55°. Find the distance between the foot of the tower and the object to the nearest whole number.

Solution

Tower = AC, object = A.

Distance between the foot of the tower and the object = BC.

Tan 55° = 60 / BC

Cross multiplying: BC × Tan 55° = 60

BC = 60 / Tan 55°

BC = 60 / 1.428 = 42.02 m

BC = 42 m (to the nearest whole number)

Evaluation

1. Define angle of elevation.
2. Define angle of depression.
3. State the sine rule and the cosine rule for triangle ABC.
4. In Example 1, which trigonometric ratio was used to find PR, and why?
5. In Example 2, why is the answer given as 42 m and not 42.02 m?

Assignment

From the top of a building 10 m high, the angle of depression of a stone lying on the horizontal ground is 70°. Calculate, correct to 1 decimal place, the distance of the stone from the foot of the building and the distance of the stone from the top of the building.