IGCSE Maths / Further trigonometry

The cosine rule: finding a missing angle

Edexcel International GCSE Maths (4MA1), chapter 25: Further trigonometry, specification point 4.8c. Free lesson.

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Use the cosine rule when you know all three sides and need an angle. Rearrange it to find the cosine, then use cos⁻¹ to get the angle.

Rearrange the cosine rule to find an angle

The cosine rule is usually given for finding a side. If you know all three sides, rearrange it so cos A is the subject, then use the inverse cosine to find A.

Choose the sides that go in the formula

The side opposite the angle you want is labelled a, so its square is subtracted in the numerator. The other two sides meet at that angle, so their product, doubled, goes underneath. For angle A, the opposite side is a. The sides meeting at A are b and c. If the question gives three sides and no angles, the sine rule cannot be used because there is no angle-side pair to match.

Work out an angle from three sides

For triangle ABC with BC = 5 cm, AC = 7 cm and AB = 8 cm, find angle A. The side opposite A is 5 cm, while the sides meeting at A are 7 cm and 8 cm.

Read a negative cosine correctly

A negative value for the cosine is not an error. It means the angle is obtuse, which is greater than 90°. The cosine rule handles this without any extra steps. For triangle ABC with BC = 6 cm, AC = 7 cm and AB = 11 cm, angle C is opposite the longest side, AB. cos C = (6² + 7² − 11²)/(2 × 6 × 7) ≈ −0.4286, so C = cos⁻¹(−0.4286) ≈ 115°.

Check the answer and avoid a side swap

Check that your angle size fits the triangle. The largest side faces the largest angle, and the shortest side faces the smallest angle. In the 5, 7, 8 triangle, angle A faces the shortest side, so it should be the smallest angle. For angle B when BC = 9 cm, AC = 10 cm and AB = 6 cm: cos B = (9² + 6² − 10²)/(2 × 9 × 6), so B ≈ 80.9° to 3 significant figures. Do not subtract the wrong side. For angle A in the 5, 7, 8 triangle, subtract 5², not 7². Using 5² + 8² − 7² over 2 × 5 × 8 gives cos A = 0.5 and 60°, which is the wrong angle for this triangle.

More lessons in Further trigonometry

• The sine rule: finding a missing side
• The sine rule: finding a missing angle
• The ambiguous case of the sine rule
• The cosine rule: finding a missing side
• Choosing between the sine rule and the cosine rule
• Area of a triangle using half ab sin C
• Finding a side or an angle from a given area
• Area of a segment
• The angle between a line and a plane
• Trigonometry in pyramids and cones
• Problem solving in three dimensions