A bearing is an angle measured clockwise from north, written with three figures, at the point you travel from or look from. To use bearings well, draw a north line at every point that appears in the wording.
This lesson supports choosing the sine or cosine rule and prepares you for two-stage journey diagrams. The right-angle work in right-triangle trigonometry is the base for the east and south parts used below.
What do the words in a bearing question mean?
“The bearing of B from A” means: stand at A, face north, turn clockwise until you face B. The angle you turn through is the bearing. The word after “from” names the point where the north line is drawn.
Three habits keep this straight:
- Draw a short north line at the starting point of the bearing.
- Measure clockwise from that line, never anticlockwise.
- Write the answer with three figures, for example 048°, not 48°.
How do north lines relate to each other?
All north lines are parallel. That means you can use angle facts between parallel lines: co-interior angles add to 180°, and alternate angles are equal.
Suppose the bearing of B from A is 065°. At B, the angle between the north line at B and the line BA is 180° − 65° = 115° (co-interior angles).
The bearing of A from B is the full clockwise turn to BA, which is 180° + 65° = 245°. This is the back bearing.
The shortcut: add 180° if the bearing is below 180°, subtract 180° if it is 180° or more.
Worked example
A ship leaves port P and sails 12 km on a bearing of 130° to reach Q. How far east and how far south of P is Q?
Step 1, draw: put a north line at P. Measure 130° clockwise. The ship heads south-east, because 130° lies between east (090°) and south (180°).
Step 2, find the angle from a useful axis: the angle from north is 130°, so the angle from the south line is 180° − 130° = 50°.
Step 3, east part: east = 12 × sin 130° = 12 × 0.7660 = 9.19 km. (sin 130° = sin 50°.)
Step 4, south part: south = 12 × cos 50° = 12 × 0.6428 = 7.71 km.
Check: 9.19² + 7.71² = 84.5 + 59.4 = 143.9, which is close to 12² = 144. The difference is rounding.
Answer: Q is 9.19 km east and 7.71 km south of P.
The mistake to watch for
A common error is to give the bearing of A from B as the same number as the bearing of B from A.
Mistaken answer: “The bearing of B from A is 065°, so the bearing of A from B is 065° as well.”
The student forgot that the north line at B is a new line and that the direction has reversed.
Another version is to write 115° after measuring only the angle inside the diagram, instead of the full clockwise turn. The correction is to draw north at B and measure clockwise all the way: 245°.
Check yourself
1. Write a bearing of 7° in three-figure form.
Show answer
007°. Three figures are needed, so zeros are added at the front.
2. The bearing of D from C is 300°. Find the bearing of C from D.
Show answer
300° is above 180°, so subtract: 300° − 180° = 120°.
3. A walker goes 20 km from X on a bearing of 210°. How far south and how far west of X is the walker?
Show answer
The angle from the south line is 210° − 180° = 30°. South = 20 × cos 30° = 17.3 km. West = 20 × sin 30° = 10 km. Check: 17.32² + 10² = 400. So the walker is 17.3 km south and 10.0 km west of X.
Where this leads next
With bearings secure, try resolving a two-stage journey. The triangle and bearings reasoning board lets you place north lines and compare them with your own sketch, and the non-calculator working trainer supports the arithmetic.
Bearing questions combine drawing, parallel-line facts and trigonometry in one go, so slips are easy to make. A teacher in one-to-one Mathematics tuition can find which of those three is causing them.