Sine rule formula: a/sin A = b/sin B
The sine rule formula is a/sin A = b/sin B = c/sin C.
What each letter means
- \(a,\ b,\ c\) the lengths of the sides
- \(A,\ B,\ C\) the angles opposite those sides (angle A is opposite side a)
When to use it
In a triangle without a right angle, when you know a side and the angle opposite it, plus one more side or angle. To find an angle, use it upside down: sin A / a = sin B / b.
Worked example
In triangle \(ABC\), \(a=7\) cm, \(b=9\) cm and \(A=40^\circ\). Find the two possible values of \(B\), to 1 decimal place.
- \(\sin B=\dfrac{9\sin40^\circ}{7}=0.8264\ldots\)
- \(B=55.7^\circ\) or \(B=180^\circ-55.7^\circ\); both leave room for \(C\) since \(40^\circ+124.3^\circ<180^\circ\)
Answer: \(B=55.7^\circ\) or \(B=124.3^\circ\)
Common mistake
Pairing a side with the wrong angle. Each side goes with the angle opposite it, not an angle next to it.
On your course
| Course | In the exam |
|---|---|
| Edexcel A Level (9MA0) | Not in the Edexcel booklet: learn it |
| Edexcel International A Level | Not in the booklet: learn it |
| 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 sine rule formula?
The sine rule formula is a/sin A = b/sin B = c/sin C. a, b, c: the lengths of the sides; A, B, C: the angles opposite those sides (angle A is opposite side a).
Is the sine rule given in the exam?
Edexcel A Level (9MA0): not in the Edexcel booklet: learn it. Edexcel International A Level: not in the booklet: learn it. 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.
What is the ambiguous case of the sine rule?
When you find an angle from two sides and an angle that is not between them, there can be two answers: θ and 180° − θ. Both are possible if the second still leaves room for the third angle.
Our own wording, examples and card, checked by A Level Math Revision. Not produced or endorsed by Pearson Edexcel, AQA or OCR.