Bearings and Navigation (AI SL)
Bearings questions turn a real-world journey - ships, planes, hikers - into a triangle you can solve. Once the diagram is drawn correctly from the three-digit bearing convention, most problems reduce to Pythagoras' theorem or the cosine rule. This page is part of the broader Triangle Trigonometry topic.
11 questions on this sub-topic.
The two formulas
Covered under IB syllabus reference SL3.3: applications of right and non-right angled trigonometry in contexts that may use bearings, including constructing a labelled diagram from a written statement. Bearings themselves are a convention rather than a formula - the cosine rule is usually what actually solves the triangle once the diagram is drawn.
Bearings
Directions measured clockwise from north, given as three digits. Two bearings from a common point define an angle you can feed into the cosine rule.
Convention, not a formulaCosine rule for a side
\(c^2=a^2+b^2-2ab\cos C\)
Once the angle between two legs of the journey is known, this gives the direct distance between the start and end points.
✓ In the formula bookletNeed the wider syllabus context and GDC settings for trig? See Triangle Trigonometry's GDC guidance.
Worked examples
Two ships leave port \(P\) at the same time. Ship \(A\) travels 15 km on a bearing of \(040^\circ\) and ship \(B\) travels 22 km on a bearing of \(110^\circ\).
Find the distance \(AB\), to 3 significant figures.
Worked solution
The angle between the two bearings is \(110^\circ-40^\circ=70^\circ.\) A1
\(AB^2=15^2+22^2-2(15)(22)\cos70^\circ.\) M1
\(=225+484-660\cos70^\circ\approx 709-225.8=483.2\Rightarrow AB\) A1 \(\approx 22.0\) km. A1
A ship sails \(12\) km on bearing \(050^\circ\), then \(8\) km on bearing \(140^\circ\) (a right-angle turn). Find the direct distance from start, to 3 significant figures.
Worked solution
\(\sqrt{12^2+8^2}\) M1
\(\approx14.4\) km. A1
Common mistakes
- Measuring a bearing anticlockwise, or forgetting the three-digit convention. Bearings are always clockwise from north and written with three digits, e.g. \(060^\circ\) not \(60^\circ\).
- Finding the wrong angle at the vertex. The angle between two bearings from the same point is their difference (or 360 minus that difference if it exceeds 180) - not one of the bearings on its own.
- Assuming every bearings question needs the cosine rule. A right-angle turn between two legs is a Pythagoras problem in disguise - check the difference between the bearings before reaching for the cosine rule.
- Skipping the sketch with a north line at each point. Bearings problems are much easier to set up correctly once a north arrow and the angle between the bearings are drawn at every vertex - without one, it's easy to compute the supplementary angle by mistake.
Ready to practise properly?
11 bearings and navigation questions, marked instantly like the real exam.
Quick answers
How are bearings written and measured?
A bearing is an angle measured clockwise from north, always given as three digits, e.g. \(060^\circ\) or \(245^\circ\), never \(60^\circ\) or \(245^\circ\) written as a plain two-digit angle.
Do bearings questions always need the cosine rule?
Not always. If a route includes a right-angle turn, Pythagoras' theorem and basic trigonometry often work directly; the cosine and sine rules come in once the angle between two legs isn't \(90^\circ\).