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 : $\mathbf{65.4^\circ}$.

Extension: useful background, not examined in AI SL. 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 : $C = \cos^{-1}(-0.265) = \mathbf{105.4^\circ}$.

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IB Maths AI SL · Unit 3: Geometry and Trigonometry

IB Maths AI SL Non-Right-Angled Trigonometry Questions

Updated October 2026

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, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

42 questions
Paper 1: 10
Paper 2: 32
9 easy
17 medium
12 hard
4 very hard
5 worked examples

Practise Non-Right-Angled Trigonometry questions →

What you need to know

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 →

What's examined in AI SL non-right-angled trigonometry

The question bank covers these non-right-angled trigonometry question types (number of questions in brackets):

Sine Rule applications (15)
Cosine Rule applications (14)
Area formula applications (13)

Key formulas

Cosine rule: $c^{2} = a^{2} + b^{2} - 2 a b \cos C$
Sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

In the same notation as the IB formula booklet. All AI SL formulas →

Explained, with a worked example: Area of a triangle · Cosine rule · Sine rule

Where students lose marks on non-right-angled trigonometry

Examiner's noteTriangles and 3D: wrong rule, wrong angle, missing units

Where marks are lost: Candidates assumed a right angle that was not there (or used the sine rule in a right-angled triangle), found an angle other than the one named, used the wrong radius or only one of two areas, missed an arc length in a curved surface, forgot units where they were required, and did not give an answer in standard form when asked. Drawing a diagram helped.

Examiners flagged this in 3 exam series (May 2024, November 2024, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Non-Right-Angled Trigonometry worked examples

Worked example 1: Side using the cosine rule · easy

In $△ A B C$ : $A B = 8$ , $A C = 11$ , $B \hat{A} C = 42^{\circ}$ . Find $B C$ to 3 s.f.

Solution

3 steps

Examiner tip: Check GDC is in DEGREES mode.

Worked example 2: Angle using the sine rule · medium

In $△ P Q R$ : $P Q = 14$ , $Q R = 18$ , $P \hat{R} Q = 45^{\circ}$ . Find the acute $R \hat{P} Q$ to 1 d.p.

Solution

3 steps

Examiner tip: For angles, keep sine terms in the numerators — easier to rearrange.

Worked example 3: The ambiguous case of the sine rule (extension) · hard

Extension: useful background, not examined in AI SL.

In $△ D E F$ : $D E = 12$ , $E F = 9$ , $E \hat{D} F = 35^{\circ}$ . Given $E \hat{F} D$ is obtuse, find it to 1 d.p.

Solution

3 steps

Examiner tip: SSA triggers the ambiguous case. The calculator gives only the acute answer. Extension: the AI SL guide lists the sine rule without the ambiguous case, so AI papers do not examine it, but it explains why a question may tell you the angle is obtuse.

Worked example 4: Area using ½ab sin C · easy

In $△ X Y Z$ : $X Y = 12$ , $X Z = 9$ , $Y \hat{X} Z = 75^{\circ}$ . Find the area to 3 s.f.

Solution

2 steps

Examiner tip:

\frac{1}{2} a b \sin C

needs the angle INCLUDED between the two known sides (SAS).

Worked example 5: Largest angle using cosine rule · medium

Triangle sides $15$ , $20$ , $28 m$ . Find the largest angle to 1 d.p.

Solution

3 steps · 1 to fill in

Examiner tip: Negative cosine

\Rightarrow

obtuse — expected for the largest angle.

Try these IB Maths AI SL non-right-angled trigonometry questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 2 marks · Paper 1

Find the exact area of a triangle with sides $6$ cm and $10$ cm, and an included angle of $30^{\circ}$ .

Attempt it and see the mark scheme →

Question 2 · medium · 3 marks · Paper 1

A triangular field has sides of length $40$ m, $50$ m, and $60$ m. Calculate the size of the largest angle in the field.

Full worked solution Free, no sign-in

The largest angle $θ$ is opposite the longest side, $60$ m (M1)
Substitute into the cosine rule: $\cos θ = \frac{40^{2} + 50^{2} - 60^{2}}{2 (40) (50)} = \frac{500}{4000} = 0.125$ M1
$θ = \cos^{- 1} (0.125) = {82.8}^{\circ}$ (3 s.f.) A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 4 marks · Paper 2

The area of triangle $D E F$ is $45 {cm}^{2}$ . Side $d = 10$ cm and side $e = 12$ cm, and the included angle $F$ is acute.

Find the size of angle $F$ . [2 marks]
Find the length of side $f$ . [2 marks]

Attempt it and see the mark scheme →

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All 42 non-right-angled trigonometry questions with mark schemes →

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FAQ

How many IB Maths AI SL non-right-angled trigonometry questions are there?

There are 42 exam-style non-right-angled trigonometry questions in the AI SL question bank (Paper 1: 10 · Paper 2: 32), graded 17 medium, 9 easy, 12 hard, 4 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is non-right-angled trigonometry on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 10 · Paper 2: 32. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

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.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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Free ready-to-teach lessons (slides and teacher notes): Sine rule · Cosine rule · Area = ½ ab sin C.

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IB Maths AI SL Non-Right-Angled Trigonometry Questions

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What you need to know

What's examined in AI SL non-right-angled trigonometry

Key formulas

Where students lose marks on non-right-angled trigonometry

Non-Right-Angled Trigonometry worked examples

Worked example 1: Side using the cosine rule · easy

Solution

Worked example 2: Angle using the sine rule · medium

Solution

Worked example 3: The ambiguous case of the sine rule (extension) · hard

Solution

Worked example 4: Area using ½ab sin C · easy

Solution

Worked example 5: Largest angle using cosine rule · medium

Solution

Try these IB Maths AI SL non-right-angled trigonometry questions

Question 1 · easy · 2 marks · Paper 1

Question 2 · medium · 3 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 4 marks · Paper 2

Revise non-right-angled trigonometry for the exam

FAQ

What else is free here

Teaching this topic?

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Non-Right-Angled Trigonometry in other IB Maths courses