IB Maths AI SL · Unit 3: Geometry and Trigonometry
Exam-style IB Maths AI SL three-figure bearings and navigation questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
Three-figure bearings measured clockwise from north. SL AI ships, planes, and hikers — always requiring a right-angled triangle plus the sine or cosine rule. Bearings and navigation problems overview →
The question bank covers these three-figure bearings and navigation question types (number of questions in brackets):
Town $B$ is $30\text{ km}$ East and $40\text{ km}$ North of Town $A$. Find the three-figure bearing of $B$ from $A$.
1. Ratio: $\tan\theta = \frac{30}{40} = 0.75$.
2. Compute: $\theta = 36.87^\circ$.
3. Three-figure: $\mathbf{037^\circ}$.
Ship $P$ on bearing $050^\circ$ for $12\text{ km}$; ship $Q$ on bearing $130^\circ$ for $15\text{ km}$. Find $PQ$.
1. Included angle: $130 - 50 = 80^\circ$.
2. Cosine rule: $d^2 = 144 + 225 - 360\cos 80^\circ = 306.48\ldots$
3. State: $d = \mathbf{17.5\text{ km}}$.
Hiker: $5\text{ km}$ on $060^\circ$ from $B$ to $P$, then $8\text{ km}$ on $150^\circ$ from $P$ to $C$. Find the bearing from $C$ back to $B$.
1. Angle at $P$: back-bearing $240^\circ - 150^\circ = 90^\circ$ (right angle).
2. Angle at $C$: $\tan C = \frac{5}{8} \implies C = 32.0^\circ$.
3. Back-bearing of leg 2: $150 + 180 = 330^\circ$.
4. Return bearing: $330 - 32 = \mathbf{298^\circ}$.
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
A ship sails from a port, \(P\), travelling \(40 \text{ km}\) on a bearing of \(060^{\circ}\) to reach a marker, \(M\). At the marker, the ship changes direction and sails \(30 \text{ km}\) on a bearing of \(150^{\circ}\) to reach an island, \(I\).
A boat leaves a port \(P\) and sails \(20 \text{ km}\) on a bearing of \(040^{\circ}\) to a buoy \(B\). It then turns and sails \(35 \text{ km}\) on a bearing of \(140^{\circ}\) to an island \(I\).
Two coastguard vessels, Alpha and Beta, leave the same dock at 12:00. Alpha travels at a speed of \(15 \text{ km/h}\) on a bearing of \(050^{\circ}\). Beta travels at a speed of \(22 \text{ km/h}\) on a bearing of \(125^{\circ}\).
All 10 three-figure bearings and navigation questions with mark schemes →
There are 10 exam-style three-figure bearings and navigation questions in the AI SL question bank (Paper 1: 3 · Paper 2: 7), graded 2 easy, 2 medium, 4 hard, 2 very hard. Every question has a full IB-style mark scheme (M, A and R marks).
Both. In the bank, Paper 1: 3 · Paper 2: 7. Practise with your GDC — AI papers expect calculator methods throughout.
Open the AI SL Unit 3 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.