[ Law of Sines ]

Express the area of triangle $ABC$ in terms of the length of side $BC$ and the measures of angles $B$ and $C$.

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According to the Law of Sines, $b=a \sin \beta / \sin \alpha=a \sin \beta / \sin (\beta+\gamma)$, therefore $S=a b \sin \gamma / 2=a^{2} \sin \beta \sin \gamma / 2 \sin (\beta+\gamma)$.