The sine rule

Higher only

The idea

In any triangle, $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ , with each side opposite its angle. Use it when you know a side and the angle opposite it.

Label sides with small letters opposite their capital angles.
For a side, use $\frac{a}{\sin A} = \frac{b}{\sin B}$ ; for an angle, flip it: $\frac{\sin A}{a} = \frac{\sin B}{b}$ .
When finding an angle, $180^\circ$ minus your answer may also work.

Watch out An obtuse angle can share a sine with an acute one: check both.

See it worked

Worked exampleCalculator allowed2 marks

In triangle $ABC$ , angle $A = 40^\circ$ , angle $B = 65^\circ$ and $BC = 8$ cm.

Work out the length of $AC$ . Give your answer correct to 3 significant figures.

$BC$ is opposite angle $A$ and $AC$ is opposite angle $B$ .
Sine rule. $\frac{AC}{\sin 65^\circ} = \frac{8}{\sin 40^\circ}$
$AC = \frac{8 \sin 65^\circ}{\sin 40^\circ} = 11.279\ldots$

Answer 11.3 cm

Your turn

Question 1Calculator allowed2 marks

In triangle ABC, angle A is $34^\circ$ , angle B is $79^\circ$ and BC = 8 cm. Work out AC. Give your answer to 3 significant figures.

Question 2Calculator allowed4 marks

In triangle ABC, angle A is $34^\circ$ , angle B is $71^\circ$ and side BC is 9 cm.

(a) Find AC to 3 significant figures.

(b) Find angle C.

Question 3Calculator allowed3 marks

In triangle ABC, AB = 12 cm, AC = 9 cm and angle ABC = $35^\circ.$ There are two possible triangles. Work out the larger possible area. Give your answer to 3 significant figures. You must show your working.

In past papers

35 official questions in 35 papers since 2017. The latest:

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