IGCSE Maths / Further trigonometry

The cosine rule: finding a missing side

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

Read the revision notes · IGCSE Maths past papers

The cosine rule finds a missing side when you know two sides and the angle between them. It is useful when the sine rule cannot be started, because you do not have a known side and its opposite angle.

When to use the cosine rule

Use the cosine rule when you know two sides and the angle between them, but do not have a side with its opposite angle. The sine rule needs that opposite pair, so it cannot start in this situation. The angle between the two known sides is opposite the missing side.

Match the letters to the triangle

The rule is similar to Pythagoras, with a correction term. If A is 90°, cos A is 0, so the correction disappears and the rule becomes Pythagoras’ theorem. a is the side you want, opposite angle A. b and c are the sides that meet at angle A. Angles are measured in degrees.

Find a missing side with an acute angle

For triangle ABC, AC = 7 cm and AB = 10 cm, with angle A = 55°. The missing side BC is opposite A, so set a = BC. Keep the full calculator value until the final answer, since rounding early can lose the accuracy mark.

Keep the correction term together

In the rule, 2bc cos A is one lump. Multiply 2bc by cos A first, then subtract the result from b² + c². Subtracting 2bc before multiplying changes the calculation and can give an impossible side length. For the 7 cm, 10 cm, 55° example, the wrong order gives (49 + 100 − 140) × cos 55° = 9 cos 55° ≈ 5.16, so the side would be about 2.27 cm. A side of 2.27 cm is shorter than both given sides, which cannot make this triangle.

Use the rule with an obtuse angle

The method does not change when the angle is greater than 90°. For an obtuse angle, its cosine is negative. The minus sign in the rule therefore makes the correction term add to the sum of the squared sides. Enter the angle directly into your calculator and let it handle the sign.

Check the method and round at the end

In triangle PQR, angle P lies between PQ = 8.1 cm and PR = 5.4 cm, so use the cosine rule to find QR. Work through the full calculation before rounding the answer to 3 significant figures.

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 angle
• 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