Sine Rule (Proof)

Proof:
From the Trigonometric Formula for Areas of Triangles,
\[\frac{1}{2}bc\sin A = \frac{1}{2}ac\sin B = \frac{1}{2}ab\sin C\]
\[bc\sin A = ac\sin B = ab\sin C\]
\[\frac{{bc\sin A}}{{abc}} = \frac{{ac\sin B}}{{abc}} = \frac{{ab\sin C}}{{abc}}\]
\[\frac{{\sin A}}{a} = \frac{{\sin B}}{b} = \frac{{\sin C}}{c}\]
Thus,
\[\frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin C}}\]
where \(\angle A\) is the angle opposite the side \(a\), \(\angle B\) is the angle opposite the side \(b\) and \(\angle C\) is the angle opposite the side \(c\).