The Cosine Rule

Grade 6

When to use the cosine rule

The cosine rule handles the cases the sine rule can't: two sides and the included angle (SAS), or all three sides (SSS). The included angle is the one between the two known sides.

a² = b² + c² − 2bc cos A

To find an angle when you know all three sides, rearrange:

cos A = (b² + c² − a²) ÷ (2bc)

A link to Pythagoras

Notice that when A = 90°, cos A = 0, and the formula becomes a² = b² + c². The cosine rule is actually a generalisation of Pythagoras' theorem.

SAS → find a side (direct version). SSS → find an angle (rearranged version). Make this your decision tree and you will never get confused about which form to use.

Worked example

EXAMPLE

Triangle ABC has b = 7 cm, c = 10 cm, and angle A = 55°. Find side a.

a² = 7² + 10² − 2 × 7 × 10 × cos 55° = 49 + 100 − 140 × 0.5736 = 149 − 80.3 = 68.7. a = √68.7 ≈ 8.29 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?