The Sine Rule, Cosine Rule and Triangle Area

Starting point: Years 9 and 10

Label a triangle with capital letters $A$, $B$, $C$ for the angles and small letters $a$, $b$, $c$ for the sides. Side $a$ is opposite angle $A$, and so on.

The sine rule

$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$

Use it when you know a side and its opposite angle, plus one more side or angle.

The cosine rule

$$c^2=a^2+b^2-2ab\cos C$$

Use it when you know two sides and the angle between them (to find the third side), or all three sides (to find an angle). For an angle, rearrange:

$$\cos C=\frac{a^2+b^2-c^2}{2ab}$$

Area of any triangle

$$\text{Area}=\frac{1}{2}ab\sin C$$

Two sides and the angle between them. For the triangle above, area $=\frac{1}{2}(7)(9)\sin 40^{\circ }\approx 20.2{\text{ cm}}^2$.

Quick summary

• Side $a$ is opposite angle $A$.
• Sine rule: $\frac{a}{\sin A}=\frac{b}{\sin B}$. Needs a side with its opposite angle.
• Cosine rule: $c^2=a^2+b^2-2ab\cos C$. Two sides and the angle between, or three sides.
• Area $=\frac{1}{2}ab\sin C$.
• Finding an angle with the sine rule may give two answers: $\theta$ and $180^{\circ }-\theta$.

Quiz