Sine rule

#### Tier: #Higher

Description

The sine rule applies to any triangle (not just right-angled). Use it when you know two angles and a side, or two sides and a non-included angle.

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

Or, rearranged to find an angle: $$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$

Example — finding a side: In a triangle with $A = 40°$, $B = 70°$, $a = 8\text{ cm}$: $$b = \frac{8 \times \sin 70°}{\sin 40°} \approx \frac{8 \times 0.940}{0.643} \approx 11.7\text{ cm}$$

The ambiguous case: when finding an angle using the sine rule, there may be two possible solutions (since $\sin \theta = \sin(180° - \theta)$). Always check whether the second solution is valid.

Common error: using the formula with the included angle (use cosine rule instead), or forgetting the ambiguous case when finding angles.

Links

Substitution Cosine rule Bearings Area of a triangle with no right angle

Store intermediate trig values in memory (M+) to avoid rounding errors mid-calculation.