GCSE Maths revision · Higher: geometry

The cosine rule

Higher only

The idea

$a^2 = b^2 + c^2 - 2bc\cos A$ . Use it when you know two sides and the angle between them, or all three sides.

Find a missing side: substitute and square root.
Find an angle: $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$ .
The angle $A$ is opposite side $a$ .

Watch out Work out $b^2 + c^2 - 2bc\cos A$ in one go on the calculator; do not subtract $2bc$ first.

See it worked

Worked exampleCalculator allowed3 marks

The diagram shows triangle $ABC$ .

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

Two sides and the angle between them: cosine rule.
$BC^2 = 7^2 + 9^2 - 2 \times 7 \times 9 \cos 52^\circ = 52.426654\ldots$
$BC = 7.240625\ldots \approx 7.24\text{ cm (3 s.f.)}$

Answer 7.24 cm

Your turn

Question 1Calculator allowed3 marks

The diagram shows a triangle with sides of 10 cm and 8 cm enclosing an obtuse angle $\theta$ .

The area of the triangle is 30 cm².

Work out the length of the third side.

Give your answer correct to 3 significant figures.

You must show your working.

Question 2Calculator allowed4 marks

The diagram shows a triangular garden.

(a) Work out the length of the third side.

Give your answer correct to 3 significant figures.

(b) Work out the area of the garden.

Give your answer correct to 3 significant figures.

Question 3Calculator allowed3 marks

From harbour A, boat B is 12 km away on a bearing of $060^\circ$ and boat C is 9 km away on a bearing of $140^\circ.$ Work out the bearing of C from B. Give your answer as a three-figure bearing to the nearest degree. You must show your working.

In past papers

32 official questions in 32 papers since 2017. The latest:

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