GCSE & IGCSE maths practice

Cosine rule questions and answers

The cosine rule works in any triangle: $a^2 = b^2 + c^2 - 2bc \cos A$ . Use it when you know two sides and the angle between them (to find the third side), or all three sides (to find an angle, with $\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}$ ). A negative cosine means the angle is obtuse.

8 questions · Higher tier · Every answer is checked as you type it, with a full worked solution. Updated 27 Oct 2026.

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Where cosine rule comes up

GCSE (AQA, Edexcel, OCR)	Higher only: know and apply the cosine rule to find unknown lengths and angles.
Cambridge IGCSE (0580)	Extended only: use the sine and cosine rules for any triangle, including problems with obtuse angles (the formulas are given).

The method

Two sides and the included angle → find the third side.
Three sides → find an angle: cos A = (b² + c² − a²) ÷ 2bc.
Side a is opposite angle A.
Work out 2bc cos A before subtracting.

Cosine rule questions

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Warm-up

Question 1Warm-upHigher

In triangle ABC, $b = 5$ cm, $c = 7$ cm and $A = 60°$ . Find $a$ , to 3 significant figures.

Question 2Warm-upHigher

In triangle ABC, $b = 8$ cm, $c = 11$ cm and $A = 40°$ . Find $a$ , to 3 significant figures.

Exam-style

Question 3Exam-styleHigher

A triangle has sides 7 cm, 8 cm and 9 cm. Find its largest angle, to 1 decimal place.

Question 4Exam-styleHigher

A triangle has sides 5 cm, 6 cm and 10 cm. Find its largest angle, to 1 decimal place.

Question 5Exam-styleHigher

In triangle ABC, $b = 4$ cm, $c = 6$ cm and $A = 120°$ . Find $a$ , to 3 significant figures.

Question 6Exam-styleHigher

You know all three sides of a triangle and want an angle. Which should you use?

Challenge

Question 7ChallengeHigher

A boat sails 5 km due north, then 8 km on a bearing of $060°$ . How far is it from its start, to 3 significant figures?

Question 8ChallengeHigher

Two sides of a triangle are 6 cm and 9 cm, with an angle of $35°$ between them. Find the third side, to 3 significant figures.

Common mistakes

Calculating (b² + c² − 2bc) × cos A instead of subtracting 2bc cos A.
Forgetting the square root at the end.
Putting the wrong side as a when finding an angle.

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Sources

The questions are our own. The “where it comes up” notes are from each board's published specification, checked on 26 October 2026:

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