The Sine Rule

Right-angled trigonometry only works in a right-angled triangle. The sine rule breaks that limit: it works in any triangle, right-angled or not. The trick is to label the triangle so each side is named after the angle opposite it — side a is opposite angle A, side b opposite B, and side c opposite C.

Then every side, divided by the sine of its opposite angle, gives the same number:

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

Use it whenever you know an angle and its opposite side, plus one more angle or side — the rule ties the known pair to the unknown one.

In any triangle, with each side opposite the angle of the same letter:

the ratios are all equal: \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C};
to find a side, use the side ÷ sin(opposite angle) form, e.g. b = \dfrac{a\,\sin B}{\sin A};
to find an angle, flip it upside down to \dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c};
and remember the angles of the triangle still sum to 180^\circ, so a missing angle often comes for free.

Labelling the triangle

Here is a scalene (no right angle, all sides different) triangle. Step through it to see how the sides a, b, c sit opposite the angles A, B, C.