Law of Cosines/Proof 3

Theorem
Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.

Then $c^2 = a^2 + b^2 - 2ab \cos C$.

Lemma
Pythagoras's Theorem is a special case of the Law of Cosines with angles $\angle A +\angle B=\angle C$ and $\angle C=\frac \pi 2$.

Proof of Lemma
Let $c^2 = a^2 + b^2$.
 * Let $\angle B$ be the angle opposite $b$

Then:

Thus $b^2 = c^2 + a^2 - 2ac \cos B$.