Sine rule, cosine rule and area of a triangle

Maths · Geometry and measures. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

These rules work in any triangle, not only right-angled ones. Label the triangle so that side a is opposite angle A, side b is opposite angle B and side c is opposite angle C.

Choose the rule from what you are given. The sine rule needs a side and the angle opposite it, plus one more side or angle. The cosine rule is for two sides and the angle between them (to find the third side), or for all three sides (to find an angle).

The area formula also needs two sides and the angle between them. If you know two angles, find the third first using the fact that the angles of a triangle add up to 180°.

Rules to remember

Sine rule: a ÷ sin A = b ÷ sin B = c ÷ sin C.
To find an angle, turn the sine rule over: sin A ÷ a = sin B ÷ b.
Cosine rule for a side: a² = b² + c² − 2bc cos A.
Cosine rule for an angle: cos A = (b² + c² − a²) ÷ (2bc).
Area of a triangle = ½ × a × b × sin C, where C is the angle between sides a and b.
A negative value of cos A means the angle is obtuse (between 90° and 180°).

Common mistake

In the cosine rule, working out b² + c² − 2bc first and then multiplying the result by cos A. Only the 2bc is multiplied by cos A, so work out 2bc cos A on its own and then subtract it.

Worked example

A triangle has sides of 3 cm and 8 cm with an angle of 60° between them. Find the length of the third side.

Two sides and the angle between them, so use the cosine rule: a² = 3² + 8² − 2 × 3 × 8 × cos 60°.
cos 60° = 1/2, so 2 × 3 × 8 × cos 60° = 48 × 1/2 = 24.
a² = 9 + 64 − 24 = 49.
a = √49 = 7 cm.

Answer: 7 cm

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