Bearings · Elevation, depression & bearings

A bearing gives a direction as an angle. Everyone agrees on where to start (north) and which way to turn (clockwise), so a bearing means the same thing to everyone.

BearingsDrag B

059°bearing of B from A

Three things make a bearing: measure at the point you're going from (A), start from north, and turn clockwise. Write it with three figures: 059°, not 59°.

The bearing of B from ANorth line at A, then clockwise: 065°

Back bearings

The bearing of A from B is the back bearing. The north lines at A and B are parallel, so the two bearings always differ by $180^\circ$ :

if the bearing is less than $180^\circ$ , add $180^\circ$ ;
if it is $180^\circ$ or more, take away $180^\circ$ .

Back bearingParallel north lines: 060° and 240° differ by 180°

Compass directions

Some questions use the older form: start from N or S, then turn towards E or W by the angle given.

Compass	Bearing
N $40^\circ$ E	$040^\circ$
S $40^\circ$ E	$180^\circ - 40^\circ = 140^\circ$
S $40^\circ$ W	$180^\circ + 40^\circ = 220^\circ$
N $40^\circ$ W	$360^\circ - 40^\circ = 320^\circ$

Turn on “Compass form” on the board to check any direction.

Compass directionsMeasured from north or south, towards east or west

More: compass directions and turning

How far north, how far east

A journey of $d$ on a bearing $\theta$ between $000^\circ$ and $090^\circ$ is the hypotenuse of a right-angled triangle. Its other two sides say how far north and how far east you end up:

\text{north} = d\cos\theta, \qquad \text{east} = d\sin\theta

North and east partsThe bearing is the angle at the start, next to the north side

For other bearings, use the angle between the journey and the north–south line (for $150^\circ$ that’s $30^\circ$ from south), then decide the directions from the sketch.

More: how far north, south, east or west

A past question, step by step

Worked example · WAEC 2021

WAEC 2021 · Paper 1 · Q28

From a point $T$ , a man moves $12\text{ km}$ due west and then $12\text{ km}$ due south to another point $Q$ . Calculate the bearing of $T$ from $Q$ .

Sketch the route
Start at $T$ . Go 12 km due west, then 12 km due south, to reach $Q$ .
Think first. After 12 km west and 12 km south, where is T compared with Q?
Look back from Q
From $Q$ , the point $T$ is 12 km north and 12 km east. The triangle is right-angled and isosceles, so the line $QT$ is at $45^\circ$ to north, turning towards east.
Write the bearing
Measured at $Q$ , clockwise from north: $045^\circ$ . The answer is C. (The bearing of $Q$ from $T$ would be the back bearing, $225^\circ$ , which is option A: read carefully which point the question starts from.)

Your turn

WAEC 2024 · Paper 1 · Q24

Elevation, depression & bearings

Given that $P$ is $25\text{ m}$ on a bearing of $330^\circ$ from $Q$ , how far south of $P$ is $Q$ ?

Worked solution (try it first)

A bearing of $330^\circ$ is $30^\circ$ west of north, so $QP$ makes $30^\circ$ with the north line through $Q$ .
The distance north is adjacent to that $30^\circ$ angle: $25\cos30^\circ \approx 21.7$ m.
So $P$ is 21.7 m north of $Q$ , and $Q$ is 21.7 m south of $P$ , option B.

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