Law of Sines

The law of sines holds for any triangle (not just right triangles):

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

where $a, b, c$ are the side lengths opposite to angles $A, B, C$, and $R$ is the circumradius.

Use cases:

1. AAS or ASA: given two angles and one side, find the other sides.
2. SSA (ambiguous case): given two sides and a non-included angle. May yield zero, one, or two valid triangles — always check.

The law of cosines $c^2 = a^2 + b^2 - 2ab\cos C$ is the companion theorem for SSS and SAS cases. Together they fully solve any triangle: given any three independent pieces of information, you can find all six (3 sides + 3 angles).

Proof: drop an altitude from one vertex; it has length $b \sin A$ measured one way and $a \sin B$ measured the other way. Equate to get $a/\sin A = b/\sin B$.