Sine and cosine rules

The sine and cosine rules work in any triangle, not just right-angled ones. Label the sides a, b, c opposite angles A, B, C.

Key facts

  • Sine rule: a/sin A = b/sin B = c/sin C. Use it with matching pairs.
  • Cosine rule: a² = b² + c² − 2bc cos A. Use it with two sides and the angle between, or three sides.
  • Area = ½ab sin C.

Worked examples

Example 1

In triangle ABC, b = 8 cm, c = 11 cm and A = 50°. Find a.

  1. Two sides and the angle between them.
  2. a² = 8² + 11² − 2 × 8 × 11 × cos 50° a² = 71.87…
  3. a ≈ 8.48

Answer: a ≈ 8.48 cm

Example 2

In triangle PQR, p = 7 cm, P = 40° and Q = 65°. Find q.

  1. q/sin 65° = 7/sin 40°
  2. q = 7 sin 65° / sin 40° ≈ 9.87

Answer: q ≈ 9.87 cm