Sine Rule Proof

The proof or derivation of the rule is very simple. As shown above in the diagram, if you draw a perpendicular line OZ to divide the triangle, you essentially create two triangles XOZ and YOZ. They both share a common side OZ. By using a simple trigonometry formula, you can create two expressions for the side OZ.

The length h of side OZ derives from triangle XOZ and angle A.

You can also obtain the same length h through triangle YOZ and angle B.

Since the length h is the same in both cases, we can make this expression by equating them, which is common sense. After transposing, we get this familiar expression. If you want an expression for a, then multiply both sides by sin A. If you want an expression for b then multiply both sides by sin B.

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