Sine rule vs cosine rule — the decision tree (IB Math AI SL)
Updated October 2026
Quick answer: Use the sine rule, a/sin A = b/sin B, when you know a side and the angle opposite it, plus one more side or angle. Use the cosine rule, a² = b² + c² − 2bc cos A, when you know two sides and the angle between them, or all three sides. Both are in the formula booklet, and neither needs a right angle.
The single most common confusion in SL AI Paper 1. Sine rule when you have a matching angle-side pair; cosine rule when you have two sides + included angle. Practise until it's automatic.
Practise Sine rule vs cosine rule — the decision tree with AI marking
Open the full SL AI engine — every question in this unit is worked through with instant AI-graded feedback on your handwritten workings.
Label the triangle so that side a is opposite angle A, b is opposite B and c is opposite C.
Ask one question: do I know a side and the angle opposite it? If yes, use the sine rule. If no, use the cosine rule.
Sine rule: a/sin A = b/sin B = c/sin C. To find an angle, write it the other way up: sin A/a = sin B/b.
Cosine rule: a² = b² + c² − 2bc cos A for a side; cos A = (b² + c² − a²)/(2bc) for an angle when you know all three sides.
Area of any triangle: ½ab sin C, where C is the angle between the sides a and b.
AI SL questions put the triangle in a context such as bearings, heights or distances on a map: draw and label a diagram first, and check your GDC is in degrees.
Worked example
In triangle ABC, AB = 8 cm, AC = 11 cm and angle BAC = 40°. Find BC and the area of the triangle.
You know two sides and the angle between them, and no side with its opposite angle, so use the cosine rule.