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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.