Sine and cosine rules

For any triangle, label the sides a, b and c, opposite the angles A, B and C.

a/sin A = b/sin B = c/sin C (on the formulae sheet)

Use it when you know a side and its opposite angle (a matching pair).

Example: in triangle ABC, A = 40°, B = 65° and a = 8 cm. Find b, to 1 d.p.

1. a/sin A = b/sin B, so 8/sin 40° = b/sin 65°
2. b = 8 × sin 65° ÷ sin 40° = 11.27…
3. b = 11.3 cm

To find an angle, flip both fractions: sin A/a = sin B/b.

1. Write the sine rule with the angles on top.
2. Rearrange to find sin of the missing angle.
3. Use sin⁻¹.

Example: in triangle ABC, a = 7 cm, b = 10 cm and B = 70°. Find angle A, to 1 d.p.

1. sin A/7 = sin 70°/10
2. sin A = 7 × sin 70° ÷ 10 = 0.6577…
3. A = sin⁻¹(0.6577…) = 41.1°

a² = b² + c² − 2bc cos A (on the formulae sheet)

Use it to find a side when you know the other two sides and the included angle (the angle between them).

Example: in triangle ABC, b = 6 cm, c = 9 cm and A = 50°. Find a, to 1 d.p.

1. a² = 6² + 9² − 2 × 6 × 9 × cos 50°
2. a² = 36 + 81 − 69.42… = 47.57…
3. a = √47.57… = 6.89…
4. a = 6.9 cm

Rearranged, the cosine rule finds an angle when you know all three sides:

cos A = (b² + c² − a²) / 2bc

1. Put the side opposite the angle you want as a.
2. Work out cos A.
3. Use cos⁻¹.

Example: a triangle has sides of 5 cm, 7 cm and 9 cm. Find its largest angle, to 1 d.p.

1. The largest angle is opposite the longest side, so a = 9
2. cos A = (5² + 7² − 9²) ÷ (2 × 5 × 7) = (25 + 49 − 81) ÷ 70 = −0.1
3. A = cos⁻¹(−0.1) = 95.7°

A negative cosine means the angle is obtuse.

The area of any triangle, from two sides and the angle between them:

Area = ½ab sin C (on the formulae sheet)

Example: a triangle has sides of 8 cm and 11 cm, with an angle of 35° between them. Find its area, to 1 d.p.

1. Area = ½ × 8 × 11 × sin 35°
2. = 44 × sin 35° = 25.23…
3. 25.2 cm²