The Cosine Rule
The cosine rule links all three sides of a triangle with one of its angles: a^2 = b^2 + c^2 - 2bc x cosA, where a is the side opposite angle A. It is used on GCSE Higher papers to find a missing side when two sides and the angle between them (SAS) are known, or a missing angle when all three sides (SSS) are known.
Method
- Decide whether you have SAS (two sides and the included angle) to find the third side, or SSS (all three sides) to find an angle.
- For a missing side, substitute into a^2 = b^2 + c^2 - 2bc x cosA, using the two known sides as b and c and the angle between them as A, then square root the result.
- For a missing angle, rearrange to cosA = (b^2 + c^2 - a^2) / (2bc), where a is the side opposite the angle you want, then take inverse cosine.
- Work out the value inside the brackets first, then divide by 2bc before taking the inverse cosine, to avoid rounding errors.
- A negative value of cosA means the angle is obtuse (greater than 90 degrees); the calculator's inverse cosine gives the correct obtuse angle directly.
- Round the final answer to the accuracy given, including the correct units for a length or degrees for an angle.
Worked example
In triangle ABC, AB = 11 cm, BC = 8 cm and angle ABC = 65 degrees. Calculate the length of AC, giving your answer correct to 3 significant figures.
- Identify the two known sides AB = 11 cm and BC = 8 cm, and the included angle ABC = 65 degrees (SAS).
- Substitute into AC^2 = 11^2 + 8^2 - 2 x 11 x 8 x cos65.
- Calculate 11^2 + 8^2 = 185, and 2 x 11 x 8 = 176.
- Calculate cos65 = 0.4226, so AC^2 = 185 - (176 x 0.4226) = 185 - 74.38 = 110.62.
- Square root to find AC = sqrt(110.62) = 10.5176..., then round to 3 significant figures: AC = 10.5 cm (3 sf).
Practice questions
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Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
In triangle ABC, AB = 10 cm, AC = 7 cm and angle BAC = 80 degrees. Calculate BC, giving your answer correct to 3 significant figures.
A triangular sports pitch has two sides of length 40 m and 55 m, with an included angle of 75 degrees. (a) Calculate the length of the third side, giving your answer correct to 3 significant figures. (b) A fence is built along this third side, costing 6.20 pounds per metre. Using your unrounded answer to part (a), work out the total cost of the fence to the nearest penny.
The diagram shows quadrilateral WXYZ, made of triangle WXY and triangle WYZ joined along diagonal WY. In triangle WXY, WX = 20 m, XY = 14 m and angle WXY = 100 degrees. In triangle WYZ, YZ = 17 m and WZ = 19 m. (a) Calculate the length of WY, giving your answer correct to 3 significant figures. (b) Using your unrounded answer to part (a), calculate the size of angle YZW, giving your answer correct to 1 decimal place.
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