A bearing is a three-figure clockwise angle measured from north. GCSE Higher questions combine bearings with trigonometry by asking you to find a distance or angle in a triangle whose sides and angles come from bearing data. Draw a clear diagram first, mark the north lines and angles, then apply Pythagoras (right-angled) or the sine/cosine rule (non-right-angled).
Key facts about bearings
- All bearings are measured clockwise from north.
- Always written as three digits: 005°, 090°, 270°.
- North is 000° (or 360°); East is 090°; South is 180°; West is 270°.
How do you find a back bearing?
The back bearing (or return bearing) is the bearing from B to A when you know the bearing from A to B.
Rule:
- If the bearing from A to B is less than 180°: back bearing = bearing + 180°.
- If the bearing from A to B is 180° or more: back bearing = bearing − 180°.
Worked example 1:
The bearing from P to Q is 065°. Find the bearing from Q to P.
Since 065° < 180°: back bearing = 065° + 180° = 245°.
| Bearing A → B | Back bearing B → A |
|---|---|
| 040° | 220° |
| 160° | 340° |
| 200° | 020° |
| 310° | 130° |
The rule works because north lines at A and B are parallel, so alternate angles give a difference of 180°.
How do you find the angle inside a triangle from bearing data?
This is the most common GCSE skill. Draw north lines at each point, then use angle properties of parallel lines to find the interior angle of the triangle.
Worked example 2:
Ship A is at port. It sails on a bearing of 050° to point B (a distance of 80 km). It then sails on a bearing of 140° to point C (a distance of 60 km). Find the distance AC.
Step 1 — Draw the diagram: mark port A, then B north-east (050°), then C south-east of B (140°).
Step 2 — Find the angle at B inside triangle ABC:
- The bearing from A to B is 050°, so the line AB makes an angle of 50° with north at A.
- The bearing from B to C is 140°, so the line BC makes an angle of 140° with north at B.
- The angle between BA (back bearing from B to A = 050° + 180° = 230°) and BC (140°) measured at B is 230° − 140° = 90°.
Triangle ABC is right-angled at B!
Step 3 — Use Pythagoras: AC² = 80² + 60² = 6400 + 3600 = 10000 → AC = 100 km.
How do you apply the cosine rule to a bearings problem?
Worked example 3:
From a lighthouse L, a ship S is on a bearing of 040°, at a distance of 25 km. A second ship T is on a bearing of 115°, at a distance of 30 km. Find the distance ST.
Step 1 — The angle SLT at the lighthouse = 115° − 40° = 75°.
Step 2 — Two sides known (LS = 25 km, LT = 30 km) and the included angle = 75°. Use the cosine rule:
ST² = LS² + LT² − 2 × LS × LT × cos(75°)
ST² = 625 + 900 − 2 × 25 × 30 × cos(75°)
ST² = 1525 − 1500 × 0.2588
ST² = 1525 − 388.2 = 1136.8
ST ≈ 33.7 km (to 3 significant figures)
How do you find a bearing using the sine rule?
Once you know two sides and an opposite angle in a bearings triangle, use the sine rule to find a missing angle and convert back to a bearing.
Worked example 4:
Points A and B are 50 km apart. B is on a bearing of 070° from A. Point C is 40 km from A and 35 km from B. Find the bearing of C from A.
Step 1 — Use the cosine rule to find angle A:
cos A = (AB² + AC² − BC²) / (2 × AB × AC) = (2500 + 1600 − 1225) / (2 × 50 × 40) = 2875/4000 = 0.71875
Angle A = cos⁻¹(0.71875) ≈ 44.1°
Step 2 — Bearing of B from A = 070°. C is positioned such that angle BAC = 44.1°. Bearing of C from A = 070° − 44.1° = 025.9° ≈ 026°.
(The direction of C relative to B determines whether to add or subtract — always check your diagram.)
Frequently asked questions
Why do I need to draw a north line at every point?
The angle at each vertex of the triangle is NOT the bearing — it is the angle between two paths. North lines at each point (which are parallel) let you find interior angles using alternate angles, co-interior angles or basic subtraction. Skipping north lines is the single most common error in bearing problems.
When is the triangle right-angled in a bearings problem?
If the two bearings from a starting point differ by exactly 90°, or if the difference between one bearing and the back bearing of the other is 90°, a right angle appears inside the triangle. Always compute the interior angle before deciding which rule to use. Do not assume a right angle — prove it first.
What if the bearing gives a reflex angle inside the triangle?
This happens when ships change direction dramatically. The interior angle of the triangle is found by considering the geometry carefully: draw the diagram, mark north lines and use angle properties. If the interior angle exceeds 180°, you have drawn the wrong triangle — reconsider which path each leg represents.
Should I give bearings answers to the nearest degree?
Yes, unless the question specifies otherwise. Convert your calculated angle (usually in degrees and decimals) to the nearest whole degree before writing the three-digit bearing. For example, an angle of 25.9° gives a bearing of 026°.
Let Professor Pi guide you through bearings and trig problems — visit aitutors.me.