Law of Cosines/Proof 1

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


 * CosineRuleCartesian.png

Thus by definition of sine and cosine:


 * $A = \left({b \cos C, b \sin C}\right)$

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

Hence: