Law of Sines

a/ A = b/ B = c/ C — sides scale with the sines of their opposite angles.

The idea

Theorem (Law of sines).

In any triangle, with sides $a$, $b$, $c$ lying opposite the angles $A$, $B$, $C$, $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.$

Right-triangle trigonometry applies only to right triangles; the law of sines extends it to all of them. It says that every side stands in one shared proportion to the sine of the angle opposite it, so the largest angle faces the longest side and the smallest angle the shortest.

Because the ratio is shared, one side together with its opposite angle fixes it; after that, each known angle yields the side across from it, and each known side the sine of its opposite angle. The law therefore solves a triangle described by two angles and a side, or by two sides and an angle opposite one of them — the cases the law of cosines, which needs the angle between two known sides, does not cover.

The second case needs caution. Two angles between $0^{\circ}$ and $180^{\circ}$ can share a sine, one acute and one obtuse, so knowing $\sin A$ does not always determine $A$, and two different triangles may fit the same data.

Ways to work on it

• Walkthrough. Derive the law from a shared altitude, then solve for sides.
• Proof. See why the ratios are equal — one shared altitude, two right triangles.
• Practice. Two angles and one side: find the rest, with exact sines.
• Hardest. The extended law a/ A = 2R and the circumradius.

Not sure where to start? Take the ten-question placement test.