Sine rule and cosine rule
Pythagoras and SOH CAH TOA only work in right-angled triangles. For any other triangle, use the sine rule or cosine rule.
Notation
For a triangle ABC, let lowercase letters denote the sides opposite the corresponding vertex:
- a = BC (opposite A).
- b = CA (opposite B).
- c = AB (opposite C).
Sine rule
a / sin A = b / sin B = c / sin C.
Or equivalently sin A / a = sin B / b = sin C / c.
Use when you know:
- Two angles and any side (AAS or ASA).
- Two sides and a non-included angle (SSA — be wary of ambiguous case).
Cosine rule
a² = b² + c² − 2bc cos A.
(Cyclically: b² = a² + c² − 2ac cos B; c² = a² + b² − 2ab cos C.)
Rearranged for finding angles: cos A = (b² + c² − a²) / 2bc.
Use when you know:
- Two sides and the included angle (SAS) → find third side.
- Three sides (SSS) → find any angle.
✦Worked example— Worked examples
Sine rule (find side). In △ABC: A = 50°, B = 60°, c = 8 cm. Find a.
- Sum to 180°: C = 70°.
- a / sin 50° = 8 / sin 70°.
- a = 8 sin 50° / sin 70° ≈ 8 × 0.766 / 0.940 ≈ 6.52 cm.
Sine rule (find angle). In △ABC: a = 7, b = 9, A = 40°. Find B.
- 7 / sin 40° = 9 / sin B.
- sin B = 9 × sin 40° / 7 ≈ 9 × 0.643 / 7 ≈ 0.826.
- B = sin⁻¹(0.826) ≈ 55.7°.
Cosine rule (find side). In △ABC: b = 7, c = 5, A = 60°. Find a.
- a² = 49 + 25 − 2 × 7 × 5 × cos 60° = 74 − 70 × 0.5 = 39.
- a ≈ 6.24.
Cosine rule (find angle). Sides 5, 7, 9. Find the angle opposite 9.
- cos A = (25 + 49 − 81) / (2 × 5 × 7) = −7/70 = −0.1.
- A = cos⁻¹(−0.1) ≈ 95.7°.
Choosing which rule
- Got a SAS or SSS? → cosine rule.
- Got AAS or SSA? → sine rule.
⚠Common mistakes
- Wrong side opposite wrong angle — always label sides a, b, c opposite their vertex.
- Cosine rule sign error — the −2bc cos A flips for obtuse angles.
- Ambiguous SSA case — sometimes two valid triangles exist.
- Calculator in wrong mode — degrees for GCSE.
- Forgetting to take square root at the end of cosine rule for sides.
➜Try this— Quick check
In △ABC, A = 60°, b = 6, c = 4. Find a using cosine rule.
- a² = 36 + 16 − 2 × 6 × 4 × 0.5 = 52 − 24 = 28.
- a = √28 = 2√7 ≈ 5.29.
AI-generated · claude-opus-4-7 · v3-deep-geometry