The cosine rule
The cosine rule generalises Pythagoras' theorem. It is used when the sine rule cannot start: given two sides and the included angle (SAS) to find the third side, or given all three sides (SSS) to find an angle.
Key formula
Find a side: a² = b² + c² − 2bc cos A. Find an angle: cos A = (b² + c² − a²) / (2bc).
Finding a side (SAS)
Substitute the two known sides and the included angle, then take the square root.
Worked example
In triangle ABC, b = 7 cm, c = 5 cm, and A = 60°. Find a. Using a² = b² + c² − 2bc cos A: a² = 7² + 5² − 2(7)(5) cos 60° = 49 + 25 − 70 × 0.5 = 74 − 35 = 39. So a = √39 ≈ 6.24 cm.
Finding an angle (SSS)
If a = 8, b = 5, c = 7, then cos B = (a² + c² − b²)/(2ac) = (64 + 49 − 25)/(2×8×7) = 88/112 = 0.7857, so B ≈ 38.2°.
Choosing which form to use
Label the triangle so the side you want (or the angle you want) sits alone on the left. To find a side, put its opposite angle inside the cosine; to find an angle, rearrange the same rule into cos A = (b² + c² − a²)/(2bc). Always find the largest angle first when all three sides are known, because only the largest angle can be obtuse and its cosine may be negative. Substitute carefully with brackets around negative cosine values so signs are not lost.
Remember
- Use the cosine rule for SAS and SSS.
- If cos A is negative, angle A is obtuse.
- When A = 90°, cos A = 0 and it reduces to Pythagoras.