The sine rule
In any triangle ABC, side a is opposite angle A, side b opposite B, and side c opposite C. The sine rule states that each side is proportional to the sine of its opposite angle. It is used when we know either two angles and one side (AAS/ASA) or two sides and a non-included angle (SSA).
Key formula
Sine rule: a / sin A = b / sin B = c / sin C. To find an angle, use the inverted form sin A / a = sin B / b = sin C / c.
Finding a side
Rearrange to isolate the unknown. For example, b = a × sin B / sin A once a suitable ratio is chosen.
Worked example
In triangle ABC, A = 40°, B = 60°, and a = 8 cm. Find b. By the sine rule, b / sin B = a / sin A, so b = a × sin B / sin A = 8 × sin 60° / sin 40° = 8 × 0.8660 / 0.6428 = 6.928 / 0.6428 ≈ 10.78 cm.
The ambiguous case
When two sides and a non-included angle are given (SSA), there can be two possible triangles because sin θ = sin (180° − θ). Always check whether an obtuse answer is also valid.
A note on accuracy
Keep several decimal places for sine values during working, and round only at the final step. When solving for an angle you take an inverse sine, but remember the calculator returns only the acute value; if the side facing the unknown angle is the longest, the angle may instead be obtuse. Sketching the triangle roughly to scale helps you judge which answer is sensible. The sine rule also gives a quick check: the largest side must face the largest angle and the smallest side the smallest angle.
Remember
- Match each side with the sine of its opposite angle.
- Use the inverted form to find an unknown angle.
- SSA can give an ambiguous (two-triangle) case.