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)