Subsection B.4.2 Sine Law or Law of Sines
¶The sine law says that, if a triangle has sides of length \(a, b\) and \(c\) and the angles opposite those sides are \(\alpha\text{,}\) \(\beta\) and \(\gamma\text{,}\) then
\begin{align*}
\frac{a}{\sin \alpha} &= \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}.
\end{align*}
This rule is best understood by computing the area of the triangle using the formula \(A = \frac{1}{2}ab\sin\theta\) of Appendix A.10. Doing this three ways gives
\begin{align*}
2A &= bc \sin \alpha\\
2A &= ac \sin \beta\\
2A &= ab \sin \gamma
\end{align*}
Dividing these expressions by \(abc\) gives
\begin{align*}
\frac{2A}{abc} &= \frac{\sin \alpha}{a} = \frac{\sin\beta}{b} = \frac{\sin \gamma}{c}
\end{align*}
as required.