Cosine rule

The cosine rule is used in non-right-angled triangles when you have two sides and the included angle, or all three sides. It allows you to find the third side or a missing angle.

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Key facts to remember

1Cosine rule (finding a side): a² = b² + c² − 2bc cos A.
2Cosine rule (finding an angle): cos A = (b² + c² − a²) / (2bc).
3Use when given SAS (two sides and the included angle) or SSS (all three sides).
4The formula reduces to Pythagoras when A = 90° (cos 90° = 0).
5Always substitute carefully and use BIDMAS when evaluating.

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Formulas

Cosine rule (side)

a² = b² + c² − 2bc cos A

Cosine rule (angle)

cos A = (b² + c² − a²) / (2bc)

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Worked examples

Example 1

In triangle ABC, b = 7 cm, c = 10 cm, angle A = 50°. Find side a.

Working

a² = b² + c² − 2bc cos A
a² = 49 + 100 − 2(7)(10) cos 50°
a² = 149 − 140 × 0.6428 = 149 − 89.99 = 59.01
a = √59.01 ≈ 7.68 cm

Answera ≈ 7.68 cm

Example 2

In a triangle, sides are 5, 8 and 10 cm. Find the largest angle.

Working

Largest angle is opposite the longest side (10 cm)
cos A = (5² + 8² − 10²) / (2 × 5 × 8) = (25 + 64 − 100) / 80 = −11/80
A = cos⁻¹(−11/80) ≈ 97.9°

AnswerLargest angle ≈ 97.9°

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Common mistakes

✗Subtracting b² + c² from 2bc cos A instead of the other way around.

✗Forgetting to take the square root at the end when finding a side.

✗Mixing up which angle is A — A must be opposite the side you are finding.

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Exam tips

✓Decide between sine rule and cosine rule: cosine rule when you have SAS or SSS; sine rule otherwise.

✓If cos A is negative, angle A is obtuse — this is valid, so don't be alarmed.

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