Bearings and navigation problems (IB Math AI SL)

Key ideas

• Measure clockwise from north and always write three digits: 045°, not 45°.
• Draw a north line at every point you work from. The bearing back along the same path differs by 180°.
• North lines are parallel, so co-interior angles add to 180°: that is how you find the angle inside the triangle at the turning point.
• Then solve the triangle: right-angled trigonometry if there is a right angle, otherwise the sine or cosine rule. Keep the GDC in degrees.
• To give a final bearing, add the angle you found to the bearing you started from, and check the answer points the right way on your sketch.

Worked example

A ship sails 12 km from port P on a bearing of 060° to A, then 9 km on a bearing of 150° to B. Find the distance PB and the bearing of B from P.

1. At A, the bearing back to P is 060° + 180° = 240°. The angle between AP (240°) and AB (150°) is 240° − 150° = 90°, so angle PAB = 90°.
2. PB = √(12² + 9²) = √225 = 15 km.
3. Angle APB = tan⁻¹(9 ÷ 12) = 36.9° (3 s.f.).
4. Bearing of B from P = 060° + 36.9° = 096.9°, about 097°.

Answer: PB = 15 km, on a bearing of 097° from P.

Exam-style SL AI questions on this topic

• Three-Figure Bearings and Navigation questions — 10 questions with mark schemes, plus free worked examples
• All SL AI Geometry and Trigonometry questions by topic

Maths IA ideas that use bearings and navigation problems

• How far off do you end up? Navigating a route by bearings
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Related in Geometry and Trigonometry

• 3D geometry — volume, surface area, and packaging
• Sine rule vs cosine rule — the decision tree
• Voronoi diagrams — nearest-neighbour problems

Revise Bearings and navigation problems for the exam

• SL AI revision notes — every unit summarised, with the key results to learn
• SL AI past papers — specimen papers and timed practice papers
• Formulas for Geometry and Trigonometry — on the one-page SL AI formula sheet (free PDF), with the ones not in the IB booklet marked
• SL AI formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

TOK link: Can knowledge built into practices such as weaving, building or navigation count as mathematics?

Stretch yourself: original geometry problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S15 · Problem S71 · Problem S78 · Problem S21. All 22 →

Formulas in this topic: SOH CAH TOA. All formulas A–Z