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
In triangle \(ABC\), \(b=7.2\) m, \(c=5.5\) m and \(A=118^\circ\). Find \(a\).
- \(a^2=7.2^2+5.5^2-2(7.2)(5.5)\cos118^\circ\)
- \(a^2=82.09+37.18\ldots=119.27\ldots\)
Answer: \(a=10.9\) m (3 s.f.)
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 |
|---|---|
| AA SL | In the IB formula booklet |
| AI SL | In the IB formula booklet |
| AA HL | In the IB formula booklet |
| AI HL | In the IB formula booklet |
From our own IB Maths formula sheets, in our words. The IB gives its booklet to schools, so we checked against our own copy: your teacher has the official one. Official: IB DP Mathematics page.
Practise and revise
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?
AA SL: in the IB formula booklet. AI SL: in the IB formula booklet. AA HL: in the IB formula booklet. AI HL: in the IB formula booklet. This comes from our own IB 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 IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.