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.