Cosine Rule – GCSE Maths Revision Guide

The cosine rule is the go-to formula when the sine rule cannot help — specifically when you have two sides and the included angle, or when you know all three sides and need an angle.

What Is the Cosine Rule?

The cosine rule is a generalisation of Pythagoras' theorem that works for any triangle, whether or not it contains a right angle. In a triangle with sides a, b, c and opposite angles A, B, C, the rule gives a direct relationship between three sides and one angle.

When the angle A is 90°, cos 90° = 0, and the formula simplifies to a² = b² + c² — which is Pythagoras' theorem. So the cosine rule includes Pythagoras as a special case.

The cosine rule comes in two forms: one for finding a side (when you know two sides and the included angle) and one for finding an angle (when you know all three sides). Both forms are provided on the AQA and Edexcel formula sheets.

Key Formulas

a² = b² + c² − 2bc cos A (finding a side)

cos A = (b² + c² − a²) ÷ (2bc) (finding an angle)

Step-by-Step Method

Label the triangle: the side you want to find (or the side opposite the angle you want) is a, and the other two sides are b and c.
For finding a side: substitute into a² = b² + c² − 2bc cos A, calculate a², then square root.
For finding an angle: substitute into cos A = (b² + c² − a²) ÷ (2bc), calculate cos A, then use cos⁻¹.
If cos A is negative, the angle is obtuse (greater than 90°) — this is normal and expected.

Worked Example 1 — Foundation Level

Question: In triangle ABC, AB = 7 cm, AC = 9 cm and angle A = 60°. Find BC to 1 decimal place.

Working: Let a = BC, b = AC = 9, c = AB = 7, angle A = 60°. a² = b² + c² − 2bc cos A a² = 9² + 7² − 2(9)(7) cos 60° a² = 81 + 49 − 126 × 0.5 a² = 130 − 63 a² = 67 a = √67

Answer: BC = 8.2 cm (1 d.p.).

Worked Example 2 — Higher Level

Question: A triangle has sides 6 cm, 10 cm and 13 cm. Find the largest angle to 1 decimal place.

Working: The largest angle is opposite the longest side (13 cm). Let a = 13, b = 6, c = 10. cos A = (b² + c² − a²) ÷ (2bc) cos A = (36 + 100 − 169) ÷ (2 × 6 × 10) cos A = (136 − 169) ÷ 120 cos A = −33 ÷ 120 cos A = −0.275 A = cos⁻¹(−0.275)

Answer: The largest angle is 106.0° (1 d.p.). The negative cosine confirms the angle is obtuse.

Worked Example 3 — Exam Style

Question: Two walkers leave a campsite. Walker A walks 5 km on a bearing of 040°. Walker B walks 8 km on a bearing of 130°. Find the distance between the two walkers to 1 decimal place.

Working: The angle between the two paths at the campsite = 130° − 40° = 90°. With the included angle = 90°: d² = 5² + 8² − 2(5)(8) cos 90° d² = 25 + 64 − 80 × 0 d² = 89 d = √89

Answer: The walkers are 9.4 km apart (1 d.p.). (Since cos 90° = 0, this is just Pythagoras.)

Common Mistakes

Using the cosine rule when the sine rule would be simpler. If you have a complete angle-side pair, the sine rule is usually faster. Save the cosine rule for SAS and SSS situations.
Getting the sign wrong in the formula. The formula has a minus sign: a² = b² + c² minus 2bc cos A. A common error is writing plus instead, which gives the wrong answer.
Panicking when cos A is negative. A negative cosine simply means the angle is obtuse (between 90° and 180°). This is valid and expected in many exam questions.

Exam Tips

Copy the formula from the formula sheet and show the substitution clearly — this earns method marks.
For finding an angle, the largest angle is always opposite the longest side.
If your calculated value of cos A is outside the range −1 to 1, you have made an arithmetic error — go back and check.

Practice Questions

Q1 (Foundation): In triangle PQR, PQ = 11 cm, PR = 8 cm and angle P = 45°. Find QR to 1 d.p.

Answer: a² = 11² + 8² − 2(11)(8) cos 45° = 121 + 64 − 176 × 0.7071 = 185 − 124.45 = 60.55. a = √60.55 = 7.8 cm (1 d.p.).

Q2 (Foundation): A triangle has sides 5 cm, 7 cm and 9 cm. Find the angle opposite the 9 cm side to 1 d.p.

Answer: cos A = (5² + 7² − 9²) ÷ (2 × 5 × 7) = (25 + 49 − 81) ÷ 70 = −7 ÷ 70 = −0.1. A = cos⁻¹(−0.1) = 95.7° (1 d.p.).

Q3 (Higher): In triangle XYZ, XY = 14 cm, YZ = 10 cm and XZ = 18 cm. Find angle Y to 1 d.p.

Answer: Angle Y is opposite XZ = 18. cos Y = (14² + 10² − 18²) ÷ (2 × 14 × 10) = (196 + 100 − 324) ÷ 280 = −28 ÷ 280 = −0.1. Y = cos⁻¹(−0.1) = 95.7° (1 d.p.).

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Summary

The cosine rule a² = b² + c² − 2bc cos A finds a side when you know two sides and the included angle (SAS).
The rearranged form cos A = (b² + c² − a²) ÷ (2bc) finds an angle when you know all three sides (SSS).
A negative value of cos A means the angle is obtuse — this is normal, not an error.
The cosine rule is a generalisation of Pythagoras' theorem and reduces to it when the angle is 90°.

Cosine Rule –