Law of Cosines

For a random $\triangle ABC$ , $a$ , $b$ , and $c$ are the length of the sides of the triangle, and respectively, $\angle A$ , $\angle B$ , and $\angle C$ are the opposite angles, then the Law of Cosines points out:

c^2=a^2+b^2-2ab\cos{\angle C}

And similarly:

a^2=b^2+c^2-2bc\cos{\angle A}

b^2=a^2+c^2-2ac\cos{\angle B}

Derivation

Draw a prependicular line from point C to side $AB$ and intersect with it at point D, and we have:

c=\overline{BD}+\overline{AD}=a\cos{\angle B}+b\cos{\angle A}

Times $c$ on both side:

c^2=ac\cos{\angle B}+bc\cos{\angle A}

By the same reasoning:

a^2=ab\cos{\angle C}+ac\cos{\angle B}

b^2=ab\cos{\angle C}+bc\cos{\angle A}

\therefore ac\cos{\angle B}= a^2-ab\cos{\angle C}, \ bc\cos{\angle A}= b^2-ab\cos{\angle C}

\therefore c^2=ac\cos{\angle B}+bc\cos{\angle A}= (a^2-ab\cos{\angle C})+(b^2-ab\cos{\angle C})=a^2+b^2-2ab\cos{\angle C}

By draw the perpendicular line from different direction, similarly:

a^2=b^2+c^2-2bc\cos{\angle A}

b^2=a^2+c^2-2ac\cos{\angle B} \tag*{$\blacksquare$}