Non-right-angled triangles

• Label the triangle: side a is opposite angle A, side b opposite B, side c opposite C.
• Sine rule: a/sin A = b/sin B = c/sin C. Use it with a side and its opposite angle.
• Cosine rule: a² = b² + c² − 2bc cos A. Use it with two sides and the included angle, or with three sides.
• Area of a triangle = ½ab sin C.
• Obtuse angles: sin(180° − x) = sin x, and the cosine of an obtuse angle is negative.
• The ambiguous case: finding an angle with the sine rule can give two possible triangles.

Side over sine of its opposite angle

Use the sine rule when you know a side and the angle opposite it, plus one more side or angle. Write only the two fractions you need.

Two angles share a sine

To find an angle, flip the rule: sin B ÷ b = sin A ÷ a. Because sin x = sin(180° − x), the equation sin B = k has two answers between 0° and 180°: B and 180° − B. The obtuse answer is valid only if it leaves room for the third angle.

Two sides and the angle between them

When you know two sides and the angle between them, the sine rule cannot start, so use the cosine rule.

Three sides known

Rearrange to cos A = (b² + c² − a²) ÷ 2bc. The largest angle is opposite the longest side. A negative cosine means the angle is obtuse, and the calculator gives it directly, so there is no ambiguous case here.

½ab sin C

Use two sides and the angle between them. Working backwards from an area gives sin C, so again there may be an acute and an obtuse answer.