IB Maths AI SL · Unit 3: Geometry and Trigonometry
Exam-style IB Maths AI SL non-right-angled trigonometry questions with worked solutions. Start with the 5 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.
The single most common confusion in SL AI Paper 1. Sine rule when you have a matching angle-side pair; cosine rule when you have two sides + included angle. Practise until it's automatic. Sine rule vs cosine rule — the decision tree overview →
Standing at the top of a cliff looking down at a boat, or standing at the base of a tower looking up. SL AI always asks for BOTH the horizontal distance and the vertical height. 3D trigonometry — angle of elevation and depression overview →
The question bank covers these non-right-angled trigonometry question types (number of questions in brackets):
In $\triangle ABC$: $AB = 8$, $AC = 11$, $B\hat{A}C = 42^\circ$. Find $BC$ to 3 s.f.
1. Cosine rule: $BC^2 = 121 + 64 - 176\cos 42^\circ$.
2. Compute: $BC^2 = 54.20\ldots$
3. Root: $BC = \mathbf{7.36\text{ cm}}$.
In $\triangle PQR$: $PQ = 14$, $QR = 18$, $P\hat{R}Q = 45^\circ$. Find the acute $R\hat{P}Q$ to 1 d.p.
1. Sine rule: $\frac{\sin(R\hat{P}Q)}{18} = \frac{\sin 45^\circ}{14}$.
2. Solve: $\sin(R\hat{P}Q) = 0.9091\ldots$
3. Inverse: $\mathbf{65.4^\circ}$.
In $\triangle DEF$: $DE = 12$, $EF = 9$, $E\hat{D}F = 35^\circ$. Given $E\hat{F}D$ is obtuse, find it to 1 d.p.
1. Sine rule: $\sin(E\hat{F}D) = \frac{12 \sin 35^\circ}{9} = 0.7647\ldots$
2. Acute: $\sin^{-1}(0.7647) = 49.88^\circ$.
3. Obtuse: $180^\circ - 49.88^\circ = \mathbf{130.1^\circ}$.
In $\triangle XYZ$: $XY = 12$, $XZ = 9$, $Y\hat{X}Z = 75^\circ$. Find the area to 3 s.f.
1. Formula: $\text{Area} = \frac{1}{2}(12)(9)\sin 75^\circ = 54 \cdot 0.9659\ldots$
2. State: $\mathbf{52.2\text{ cm}^2}$.
Triangle sides $15$, $20$, $28\text{ m}$. Find the largest angle to 1 d.p.
1. Largest angle is opposite the longest side ($28$).
2. Cosine rule: $\cos C = \frac{225+400-784}{600} = -0.265$.
3. State: $C = \cos^{-1}(-0.265) = \mathbf{105.4^\circ}$.
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Find the exact area of a triangle with sides $6$ cm and $10$ cm, and an included angle of $30^\circ$.
A triangular field has sides of length $40$ m, $50$ m, and $60$ m. Calculate the size of the largest angle in the field.
The area of triangle \(DEF\) is \(45\text{ cm}^2\). Side \(d = 10\) cm and side \(e = 12\) cm, and the included angle \(F\) is acute.
All 23 non-right-angled trigonometry questions with mark schemes →
There are 23 exam-style non-right-angled trigonometry questions in the AI SL question bank (Paper 1: 10 · Paper 2: 13), graded 7 medium, 4 easy, 6 hard, 4 very hard, 2 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Both. In the bank, Paper 1: 10 · Paper 2: 13. 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.