The cosine rule

Statement

• a² = b² + c² − 2bc cos A.
• For an angle: cos A = (b² + c² − a²) / (2bc).
• Use it when you know two sides and the included angle (SAS) or all three sides (SSS).

Worked examples

1. Sides b = 5, c = 7, included angle A = 60°: a² = 25 + 49 − 2(5)(7)(0.5) = 74 − 35 = 39 → a = √39 ≈ 6.24.
2. Sides a = 8, b = 6, c = 5; find angle A: cos A = (36 + 25 − 64)/(2·6·5) = −3/60 = −0.05 → A ≈ 92.9°.
3. A right angle check: if a² = b² + c², then cos A = 0 and A = 90° — Pythagoras is the cosine rule with cos 90° = 0.

Choosing between the rules