Sine rule and cosine rule

These higher-tier rules for non-right-angled triangles appear regularly on OCR J560 Papers 2 and 3. Both formulae are given on the formula sheet — knowing when and how to use each is the key skill.

When to use which rule

Information given	Rule
Two sides + included angle → find 3rd side	Cosine rule
All three sides → find an angle	Cosine rule
Two angles + one side	Sine rule
Two sides + non-included angle	Sine rule

A useful mnemonic: Cosine rule = C-A-S (two sides and the angle between, or all sides → angle). Sine rule = two Angles + one Side, or two Sides + non-included Angle.

Sine rule

a/sin A = b/sin B = c/sin C

(Or equivalently: sin A/a = sin B/b = sin C/c for finding angles.)

Label the triangle: side a is opposite angle A, side b opposite angle B, side c opposite angle C.

Finding a side

Example: In triangle ABC, angle A = 48°, angle B = 73°, b = 12 cm. Find a.

• a/sin 48° = 12/sin 73° M1.
• a = 12 × sin 48° / sin 73° = 12 × 0.7431/0.9563 ≈ 9.32 cm A1.

Finding an angle

Example: In triangle PQR, p = 8 cm, q = 11 cm, angle P = 35°. Find angle Q.

• sin Q/11 = sin 35°/8 M1.
• sin Q = 11 × sin 35°/8 = 0.7893 M1.
• Q = sin⁻¹(0.7893) ≈ 52.0° A1.

Ambiguous case: sin⁻¹ may give two values (θ and 180°−θ). Consider both unless context rules one out.

Cosine rule

Finding a side

a² = b² + c² − 2bc cos A

Example: b = 7 cm, c = 10 cm, A = 60°. Find a.

• a² = 49 + 100 − 2(7)(10) cos 60° = 149 − 140(0.5) = 149 − 70 = 79.
• a = √79 ≈ 8.89 cm.

Finding an angle

cos A = (b² + c² − a²) / (2bc)

Example: a = 9, b = 6, c = 7. Find angle A.

• cos A = (36 + 49 − 81)/(2×6×7) = 4/84 = 1/21 ≈ 0.04762.
• A = cos⁻¹(0.04762) ≈ 87.3°.

Area formula (G23 link)

Area = ½ab sin C — when two sides and the included angle are known.

Common OCR exam mistakes

1. Using the sine rule when the cosine rule is needed (missing the included angle clue).
2. Labelling sides incorrectly — side a must be OPPOSITE angle A.
3. In cosine rule: subtracting instead of adding (b² + c² − 2bc cos A, not b² + c² + 2bc cos A).
4. Not considering the ambiguous case with the sine rule.
5. Calculator not in degree mode.