Sine and cosine rules and area of a triangleEdexcel International A Level Maths: Revision notes

For a triangle with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}.$ Use the sine rule when you know a side and its opposite angle plus one more side or angle. To find an angle, the inverted form $\frac{\sin A}{a}=\frac{\sin B}{b}$ is easier. Example: $AB=8.4$, $B=52^{\circ}$, $C=71^{\circ}$ gives $AC=\frac{8.4\sin52^{\circ}}{\sin71^{\circ}}=7.00$.

Use the cosine rule with two sides and the included angle (to find the third side) or with all three sides (to find an angle): $a^2=b^2+c^2-2bc\cos A,\qquad \cos A=\frac{b^2+c^2-a^2}{2bc}.$ The side on the left must be opposite the angle. A negative cosine means the angle is obtuse. Example: $12$, $9$ and included $40^{\circ}$ give $QR^2=225-216\cos40^{\circ}=59.5$, so $QR=7.72$.

When you are given two sides and a non-included angle (SSA), finding another angle with the sine rule can give two answers, because $\sin\theta=\sin(180^{\circ}-\theta)$. Example: $AB=12$, $BC=8$, $A=35^{\circ}$: $\sin C=\frac{12\sin35^{\circ}}{8}=0.860$, so $C=59.4^{\circ}$ or $C=120.6^{\circ}$. Both are valid if the angle sum stays below $180^{\circ}$: $35^{\circ}+120.6^{\circ}=155.6^{\circ}$, so there are two triangles. If $\sin\theta>1$ there is no triangle; if the obtuse angle makes the angle sum exceed $180^{\circ}$, only one triangle exists. The problem arises only when the given angle is opposite the shorter of the two given sides.

$\text{Area}=\tfrac12ab\sin C,$ where $C$ is the angle between sides $a$ and $b$. Example: $12$, $9$ and $40^{\circ}$ give $\frac12(12)(9)\sin40^{\circ}=34.7$. If you have not been given the included angle, find it first with the sine or cosine rule. In the ambiguous case the two triangles have different areas, so work out each: $47.9$ cm$^2$ and $19.8$ cm$^2$ in the example above.

• A side and its opposite angle known, plus one other piece of data: sine rule.
• Two sides and the included angle, or three sides: cosine rule.
• Two sides and a non-included angle: sine rule, and check for two solutions.
• Area: $\frac12ab\sin C$.

Keep full calculator values until the end and round at the final step (lengths to 3 s.f., angles to 1 d.p.). The largest side is always opposite the largest angle, which is a useful check.