The Sine Rule

Grade 6

When to use the sine rule

The sine rule works in any triangle — not just right-angled ones. Use it when you have a matching side-angle pair (one side and its opposite angle), plus one more piece of information.

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

Lowercase letters are sides; uppercase are the opposite angles. You can also flip the fractions: sin A/a = sin B/b = sin C/c — useful when finding an angle.

Setting it up

Label the triangle: call the known side–angle pair a and A. Then use b/sin B = a/sin A (or the angle version) to find the unknown.

Use the side-finding version (a/sin A = ...) when you need a length. Use the angle-finding version (sin A/a = ...) when you need an angle. Two versions, same rule.

Worked example

EXAMPLE

In triangle ABC, angle A = 42°, angle B = 73°, and side a = 9 cm. Find side b.

b/sin B = a/sin A. b = a × sin B / sin A = 9 × sin 73° / sin 42° = 9 × 0.9563 / 0.6691 ≈ 12.9 cm (to 3 s.f.).

Try this one

A student needs to find a missing side in a triangle.

They know two sides and the angle between them.

(a) Which rule should they use: the sine rule or the cosine rule? [1]

(b) Write down the cosine rule for finding side \( a \). [1]

Nova's hint:

This tests when each rule applies.

Ask yourself: 'do I have a matching side and its opposite angle (sine rule), or two sides and the angle between them (cosine rule)?' Think about which pieces of information you are given.

Try to Solve it with Nova?