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3-figure · from N

03Bearings

Key idea

A bearing is an angle measured clockwise from North, always written with three figures. Bearings problems are really trig problems in disguise: draw the North line at each point, mark the angles, then reach for right-angled trig, the sine rule or the cosine rule.

Writing bearings

Always three figures, clockwise from North: due East is 090°, due South is 180°, and 7° is written 007°.

Back bearings

The bearing of A from B differs from the bearing of B from A by exactly 180°. Add 180° if the original is under 180°, otherwise subtract.

Finding the angle inside the triangle

Draw a North line at each vertex. Use the fact that North lines are parallel — co-interior angles between them sum to 180° — to find the interior angle you need.

Measured clockwise from North
Always 3 figures: 007°, 090°, 245°
Back bearing = bearing ± 180°
Common error — Forgetting the third figure (writing 68° instead of 068°) or measuring anticlockwise. Bearings are always clockwise from North.
Worked examples
Example 1 · Back bearing

The bearing of C from A is 040°. Find the bearing of A from C.

  1. Back bearing = 040° + 180°
  2. = 220°
  3. So A is on a bearing of 220° from C.
Example 2 · Bearings + right angle

A ship sails 30 km from P on a bearing of 070° to Q, then 40 km on a bearing of 160° to R. Find the direct distance PR.

  1. Turn angle at Q = 160° − 070° = 90°, so triangle PQR is right-angled at Q.
  2. PR² = 30² + 40² = 900 + 1600 = 2500
  3. PR = √2500
  4. PR = 50 km

Exam-style question bank

5 of 5 shown
Foundations[2 marks]

The bearing of a lighthouse from a boat is 115°. Find the bearing of the boat from the lighthouse.

Foundations[2 marks]

Point Q is due South of point P. Write down the bearing of Q from P.

Standard[3 marks]

The bearing of B from A is 250°. Find the bearing of A from B.

Standard[4 marks]

A boat sails 12 km on a bearing of 040°, then turns and sails 12 km on a bearing of 130°. Calculate the direct distance from start to finish.

Challenge[6 marks]

From port A a ship sails 20 km on a bearing of 050° to B, then 35 km on a bearing of 110° to C. Calculate the distance AC.