# The Law of Sines Given any triangle $ABC$ such that the side $a$ is opposite point $A$, the side $b$ is opposite point $B$, the side $c$ is opposite point $C$, and that the angles of $ABC$ are $\alpha$, $\beta$ and $\gamma$ respectively, the following holds true: $\frac {\sin \alpha}{a} = \frac {\sin \beta}{b} = \frac {\sin \gamma}{c}$ Put differently: the ratio of the sine of an angle and the angle's opposite side is constant in every triangle. ## Proof ![[law of sines.png|400]] Given a generic triangle $ABC$ drop a height $h_1$ from $C$ and analyze its length. $h_1 = b \sin \alpha = a\sin \beta$. This can be rewritten as: $ \frac {\sin \alpha} a = \frac {\sin \beta} b $ Now drop another height $h_2$ from $A$ and analyze its length. $h_2 = c \sin \beta = b \sin \gamma$. This can be rewritten as: $ \frac {\sin \beta} b = \frac {\sin \gamma} c $