Solving Bearings and Navigation Problems

What Are Bearings?

A bearing is a way of describing direction using angles measured clockwise from north. Bearings are used in navigation, surveying, and mapping, and they appear regularly in IGCSE Maths exams on both the Core and Extended syllabuses.

Bearings follow three strict rules:

They are measured from north
They are measured clockwise
They are written as three-figure numbers (for example, 045° not 45°, and 008° not 8°)

Getting these three rules right from the start prevents the most common errors in bearing questions.

Drawing a Bearings Diagram

Almost every bearings question requires a diagram. Even when one is provided, you should add to it. Here is a reliable process:

Step 1: Draw a North Line at Every Point

At every location mentioned in the question, draw a vertical arrow pointing upward to represent north. This is critical — you measure bearings from these north lines, so every point needs one.

Step 2: Measure the Bearing Clockwise from North

Starting from the north line, measure the angle clockwise to the direction of travel. Mark this angle clearly.

Step 3: Add Distances

Label each path with its distance. If no distance is given, just sketch the directions to work out the angle relationships.

Common mistake: Drawing the bearing from the wrong point. “The bearing of B from A” means you stand at A, look north, and measure clockwise to the direction of B. The angle is at point A.

Understanding “Bearing of B from A” vs “Bearing of A from B”

This distinction catches many students. Let us be very clear:

Bearing of B from A: Stand at A, face north, turn clockwise until you face B. The angle you turn through is the bearing.
Bearing of A from B: Stand at B, face north, turn clockwise until you face A. This is a different angle.

The two bearings are related but not equal. If the bearing of B from A is θ, then the bearing of A from B is:

θ + 180° (if θ < 180°)
θ − 180° (if θ ≥ 180°)

This is called the back bearing or reverse bearing. It is frequently tested, so practise it until the calculation is automatic.

Worked Example

The bearing of town Q from town P is 065°. Find the bearing of P from Q.

Since 065° < 180°, the bearing of P from Q = 065° + 180° = 245°.

If instead the bearing of Q from P were 230°, then the bearing of P from Q = 230° − 180° = 050°.

Combining Bearings with Trigonometry

Many IGCSE bearings questions require you to find distances or angles using trigonometry. These problems typically form right-angled triangles or non-right-angled triangles that need the sine or cosine rule.

Right-Angled Triangle Problems

These often involve a journey that goes north/south and then east/west, forming a right angle.

Worked Example: A ship sails 30 km on a bearing of 060° from port P to point Q. Find how far north and how far east the ship has travelled.

The northward distance (adjacent to the 60° angle from north) = 30 × cos 60° = 15 km
The eastward distance (opposite to the 60° angle from north) = 30 × sin 60° = 25.98 km

Key insight: When the bearing is measured from north, the north-south component uses cosine and the east-west component uses sine. This is the opposite of what you might expect from standard angle measurement, because bearings are measured from the vertical (north) rather than the horizontal.

Non-Right-Angled Triangle Problems

When the journey involves three or more points forming a non-right-angled triangle, you will need the sine rule or cosine rule.

Strategy:

Draw the diagram with north lines at every point
Use parallel north lines and angle rules (co-interior angles, alternate angles) to find the angles inside the triangle
Apply the appropriate trigonometric rule

Worked Example: A ship sails from A to B on a bearing of 070° for 50 km, then from B to C on a bearing of 150° for 40 km. Find the distance AC.

At B, the angle between the north line and BA is 070° + 180° = 250° (back bearing)
The angle ABC (inside the triangle) = 360° − 250° − (360° − 150°) …

Actually, let us think about this more carefully using the north line at B:

The direction from B back to A is 070° + 180° = 250° (measured clockwise from north at B)
The direction from B to C is 150°
The angle ABC = 250° − 150° = 100°

Now use the cosine rule: AC² = 50² + 40² − 2(50)(40) cos 100° AC² = 2500 + 1600 − 4000 × (−0.1736) = 4100 + 694.4 = 4794.4 AC = 69.2 km

Finding Bearings from Calculated Angles

After finding distances using trigonometry, you often need to state a bearing. This requires converting the angle you calculated back to a three-figure bearing measured clockwise from north.

Steps:

Identify the angle your trigonometric calculation gave you
Determine which direction from north this angle corresponds to
Convert to a clockwise-from-north measurement
Write as a three-figure number

This step requires careful thought about the geometry. Always refer back to your diagram.

Bearings and Pythagoras

Some simpler bearings questions create right-angled triangles that need Pythagoras’ theorem rather than trigonometry.

Example: Town B is 8 km due east of town A. Town C is 6 km due north of town B. Find the bearing of C from A.

The distance AC = √(8² + 6²) = √(64 + 36) = √100 = 10 km
The angle at A (measured from north to AC) is found using tan: tan θ = 8/6 (east/north)…

Wait — let us set this up properly. From A, the direction to C goes east and then north. The angle from north (clockwise) to the direction AC is:

tan(angle from north) = 8/6 = 1.333… angle from north = tan⁻¹(1.333) = 53.1°

The bearing of C from A = 053° (to three figures).

Common Mistakes to Avoid

Measuring anticlockwise instead of clockwise. Bearings are always clockwise from north.
Forgetting the three-figure format. Always pad with leading zeros: 045°, not 45°.
Drawing north lines incorrectly. North must point straight up at every point.
Confusing “from” and “to.” The bearing of B from A has the angle at A, not at B.
Using sine when you need cosine (and vice versa). Remember, north-south components use cosine of the bearing angle, east-west components use sine.
Not using the back bearing correctly. Add or subtract 180°, depending on whether the original bearing is less than or greater than 180°.

Practice Tips

Draw large, clear diagrams. Small, cramped diagrams lead to errors.
Always include north lines at every point — this is non-negotiable.
Label all known distances and angles on your diagram before starting calculations.
Double-check whether the question asks for a distance or a bearing in the final answer.

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