Length of Side of Triangle as Cosines of Angles/Proof

Proof

 * Side-of-Triangle-by-Cosines.png

Let a perpendicular be dropped from $C$ to $AB$ to meet $AB$ at $D$.

There are two cases:

Foot of Perpendicular is on $AB$
See the diagram on the left.

Foot of Perpendicular is outside $AB$
See the diagram on the right.