Mensuration and calculation: know and apply the sine rule, a/sin A = b/sin B = c/sin C and cosine rule, a² = b² + c² − 2bc cos A to find unknown lengths and angles (Higher content only)

The idea

This Higher-tier topic requires you to know and apply the sine rule and cosine rule to find unknown lengths and angles in non-right-angled triangles. The sine rule states a/sin A = b/sin B = c/sin C, where a, b, c are side lengths and A, B, C are the opposite angles. Use it when you know a matching side and angle pair plus one other side or angle. The cosine rule states a² = b² + c² − 2bc cos A, and can be rearranged to find an angle: cos A = (b² + c² − a²)/(2bc). Use it when you know two sides and the included angle, or all three sides.

For example: If b = 5 cm, c = 7 cm and A = 60°, then a² = 5² + 7² − 2×5×7×cos 60° = 25 + 49 − 35 = 39, so a = √39 cm.

What earns the marks

• Correctly states the sine rule as a/sin A = b/sin B = c/sin C and uses it to find an unknown side or angle when a matching pair is known.
• Correctly states the cosine rule as a² = b² + c² − 2bc cos A and uses it to find an unknown side when two sides and the included angle are known.
• Rearranges the cosine rule to cos A = (b² + c² − a²)/(2bc) to find an unknown angle when all three sides are known.
• Substitutes values accurately into the chosen rule, including correct use of the included angle and correct order of operations.
• Solves the resulting equation to find the unknown length or angle, giving the answer to an appropriate degree of accuracy.

Common mistakes and how to put them right

• Using the sine rule when the cosine rule is required, such as when two sides and the included angle are known.

  Put it right: Use the cosine rule for this combination, and use the sine rule only when you have a matching side and opposite angle pair.

• Forgetting to square the side lengths in the cosine rule, for example writing a² = b + c − 2bc cos A. Correction: the correct form is a² = b² + c² − 2bc cos A.

• Mislabelling sides and angles so that the side does not correspond to the opposite angle.

  Put it right: Check that side a is opposite angle A, side b opposite angle B, and side c opposite angle C before substituting.

• Using the cosine rule with the wrong angle, such as using an angle that is not included between the two known sides.

  Put it right: Identify the included angle carefully before substituting.

Exam tips

• Sketch and label the triangle clearly, marking which sides and angles are known and which are required. This helps you decide whether the sine rule or cosine rule is appropriate.
• In multiple-choice questions, check each option by substituting into the correct rule; eliminate options that use the wrong rule or incorrect order of operations.
• When finding an angle with the cosine rule, rearrange to make cos A the subject before substituting, and remember to use the inverse cosine function to obtain the final angle.