Law of Cosines/Proof 1

Proof
Let $\triangle ABC$ be embedded in a Cartesian coordinate system by identifying:
 * $C := \tuple {0, 0}$
 * $B := \tuple {a, 0}$


 * CosineRuleCartesian.png

Thus by definition of sine and cosine:


 * $A = \tuple {b \cos C, b \sin C}$

By the Distance Formula:
 * $c = \sqrt {\paren {b \cos C - a}^2 + \paren {b \sin C - 0}^2}$

Hence: