The Cosine Rule
Grade 6When to use the cosine rule
The cosine rule handles the cases the sine rule can't: two sides and the included angle (SAS), or all three sides (SSS). The included angle is the one between the two known sides.
a² = b² + c² − 2bc cos A
To find an angle when you know all three sides, rearrange:
cos A = (b² + c² − a²) ÷ (2bc)
A link to Pythagoras
Notice that when A = 90°, cos A = 0, and the formula becomes a² = b² + c². The cosine rule is actually a generalisation of Pythagoras' theorem.
SAS → find a side (direct version). SSS → find an angle (rearranged version). Make this your decision tree and you will never get confused about which form to use.
Worked example
EXAMPLE
Triangle ABC has b = 7 cm, c = 10 cm, and angle A = 55°. Find side a.
a² = 7² + 10² − 2 × 7 × 10 × cos 55° = 49 + 100 − 140 × 0.5736 = 149 − 80.3 = 68.7. a = √68.7 ≈ 8.29 cm (to 3 s.f.).
