Cosine rule formula: a² = b² + c² − 2bc cos A
The cosine rule formula is a² = b² + c² − 2bc cos A, or cos A = (b² + c² − a²) / 2bc.
What each letter means
- \(a\) the side opposite angle A
- \(b,\ c\) the other two sides
- \(A\) the angle between sides b and c
When to use it
When you know two sides and the angle between them (to find the third side), or all three sides (to find an angle, using the cos A form).
Worked example
A triangle has sides 5 cm, 7 cm and 9 cm. Find its largest angle, to 1 decimal place.
- The largest angle is opposite the longest side, 9 cm
- \(\cos C=\dfrac{5^2+7^2-9^2}{2(5)(7)}=\dfrac{-7}{70}=-0.1\)
Answer: \(C=95.7^\circ\)
Common mistake
Working out b² + c² − 2bc first and then multiplying by cos A. Find 2bc cos A as one product, then subtract it.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | Not in the Edexcel booklet: learn it |
| Edexcel International A Level | In Pearson’s IAL formulae booklet |
| AQA or OCR A Level | Check your board’s formula booklet: AQA formulae booklet (PDF) · OCR A Level Maths assessment page |
From our own A Level Maths formula sheets, in our words. Official: Pearson's formulae booklet (PDF) · Pearson's IAL formulae booklet (PDF).
Practise and revise
- Practise A Level Maths practice
- Revise Trigonometry
- Revise IAL P1 trigonometry
- Print Edexcel A Level (9MA0) one-page formula sheet
Questions
What is the cosine rule formula?
The cosine rule formula is a² = b² + c² − 2bc cos A, or cos A = (b² + c² − a²) / 2bc. a: the side opposite angle A; b, c: the other two sides; A: the angle between sides b and c.
Is the cosine rule given in the exam?
Edexcel A Level (9MA0): not in the Edexcel booklet: learn it. Edexcel International A Level: in Pearson’s IAL formulae booklet. AQA or OCR A Level: check your board’s formula booklet. This comes from our own A Level Maths formula sheets; your teacher has the official booklet.
Should I use the sine rule or the cosine rule?
Use the cosine rule when you know two sides and the angle between them, or all three sides. Use the sine rule when you know a side and the angle opposite it.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.