The Sine Rule

In any triangle, each side divided by the sine of its opposite angle gives the same value: a/sin A = b/sin B = c/sin C, where side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Only two of the three ratios are needed at once, matched to whichever side and angle are missing.

Labelling matters: side a must be the side directly across from angle A, not next to it.

Use the sine rule when a matching pair, one angle and the side opposite it, is already known, alongside one more piece of information: either two angles and a side, or two sides and a non-included angle. Use the cosine rule instead when two sides and the angle between them are known (SAS), or when all three sides are known (SSS), since the sine rule cannot be used without a known angle-side pair.

If two sides and the included angle are known, that is SAS, and the cosine rule is needed instead of the sine rule.

Set up the sine rule using the known angle-side pair and the side you want to find, then rearrange to make the missing side the subject. If the given side is not opposite one of the known angles, find the third angle first using angles in a triangle summing to 180°.

With angle A = 40°, angle B = 65°, and side a = 10 cm (opposite A), find side b: b/sin 65° = 10/sin 40°, so b = 10 × sin 65° ÷ sin 40° = 14.1 cm (3 s.f.).

To find a missing angle, use the reciprocal form of the sine rule, sin A/a = sin B/b, so the unknown sine value ends up on top. Rearrange to find the sine of the angle, then use sin⁻¹ to find the angle itself.

With angle A = 45°, side a = 9 cm (opposite A), and side b = 11 cm: sin B/11 = sin 45°/9, so sin B = 11 × sin 45° ÷ 9 = 0.864, giving B = sin⁻¹(0.864) = 59.8° (1 d.p.).