Sine, Cosine Rule & Area of TrianglesCambridge IGCSE Maths: Revision notes

The sine and cosine rules apply to any triangle, not just right-angled ones. Label a triangle so that each side is lower-case and opposite its matching upper-case angle: side $a$ opposite angle $A$, side $b$ opposite angle $B$, side $c$ opposite angle $C$.

Use the sine rule when you know an angle-side matching pair, plus one more piece of information. Use the cosine rule when you know two sides and the included angle, or all three sides.

The sine rule states: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

• To find a missing side: use the rule as written, cross-multiply
• To find a missing angle: flip the rule to $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

Use the sine rule whenever you know a matching angle-side pair plus one other side or angle.

The cosine rule states: $a^2 = b^2 + c^2 - 2bc\cos A$

• Use this form to find a missing side when two sides and the included angle are known
• Rearrange to find a missing angle when all three sides are known: $\cos A = \frac{b^2+c^2-a^2}{2bc}$

When two sides and the included angle are known, use: $\text{Area} = \frac{1}{2}ab\sin C$ where $C$ is the angle between sides $a$ and $b$. This formula is given in the exam.

Sine and cosine values behave differently for obtuse angles (between 90° and 180°):

• $\sin\theta$ is positive for both $\theta$ and $180°-\theta$, so the sine rule can give two possible answers for an angle — this is the ambiguous case. Check whether the obtuse version makes sense in context (e.g. angles in a triangle must sum to 180°)
• $\cos\theta$ is negative for obtuse angles, so the cosine rule naturally identifies whether an angle is obtuse without ambiguity