Non-right-angled trigonometry · Higher notes

For a general triangle, matching opposite sides and angles is essential: side a is opposite angle A. The sine rule connects a known opposite pair to another pair. The cosine rule is useful when you know two sides and their included angle, or all three sides. The area formula uses two sides and the angle between them. Draw and label the triangle before choosing a rule, use degree mode, and avoid assuming that a triangle is right-angled unless this is stated.

Learning objectives

Select the sine rule, cosine rule or triangle area formula.
Match opposite sides and angles, and round final values only.

Key ideas

Use opposite side-angle pairs in the sine rule.

\frac a{\sin A}=\frac b{\sin B}=\frac c{\sin C}

The cosine rule uses the included angle between the two known sides.

a^2=b^2+c^2-2bc\cos A

The area formula uses the included angle.

\text{area}=\frac12ab\sin C

Use degree mode for degree angles. A sine-rule angle may have two possible values; check the triangle context.

Worked examples

Example 1

Sides of 7 cm and 9 cm enclose 60 degrees. Find the third side and area to three significant figures.

c^2=7^2+9^2-2(7)(9)\cos60^\circ=67

c=\sqrt{67}\ \text{cm}

\text{area}=\frac12(7)(9)\sin60^\circ=\frac{63\sqrt3}{4}\ \text{cm}^2

Answer

c\approx8.19\ \text{cm},\qquad\text{area}\approx27.3\ \text{cm}^2

Example 2

An 8 cm side is opposite 30 degrees. Find the side opposite 60 degrees exactly.

\frac8{\sin30^\circ}=\frac b{\sin60^\circ}

Answer

b=\frac{8\sin60^\circ}{\sin30^\circ}=8\sqrt3\ \text{cm}

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Use the sine rule, cosine rule or one-half ab sin C area formula.

AO2 · Reasoning

Justify the choice of rule and consider whether more than one triangle is possible.

AO3 · Problem solving

Combine non-right-angled trigonometry with bearings, area or a multi-triangle diagram.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

Pythagoras requires a right angle. Match each sine-rule side to its opposite angle; use the included angle for the cosine rule.

Try it yourself

Attempt each question before opening its answer.

Check 1: Side a = 8 cm is opposite A = 30°. Find side b opposite B = 60°, exactly.

Answer: 8√3 cm

b = 8 sin 60°/sin 30° = 8√3.

Check 2: A triangle has sides 3, 4 and 5 cm. Find the angle opposite the 5 cm side.

Answer: 90°

cos A = (3² + 4² - 5²)/(2 × 3 × 4) = 0.

Explain it in your own words

Explain how to label opposite side-angle pairs.
State when you would choose the cosine rule rather than the sine rule.
Write an area formula and mark the included angle in a sketch.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All higher notes

Specification source

AQA Geometry specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.