Law of cosines

In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides ${\displaystyle a,}$ ${\displaystyle b,}$ and ${\displaystyle c,}$ opposite respective angles ${\displaystyle \alpha ,}$ ${\displaystyle \beta ,}$ and ${\displaystyle \gamma }$ (see Fig. 1), the law of cosines states:

${\displaystyle {\begin{aligned}c^{2}&=a^{2}+b^{2}-2ab\cos \gamma ,\\[3mu]a^{2}&=b^{2}+c^{2}-2bc\cos \alpha ,\\[3mu]b^{2}&=a^{2}+c^{2}-2ac\cos \beta .\end{aligned}}}$

The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if ${\displaystyle \gamma }$ is a right angle then ${\displaystyle \cos \gamma =0,}$ and the law of cosines reduces to ${\displaystyle c^{2}=a^{2}+b^{2}.}$

The law of cosines is useful for solving a triangle when all three sides or two sides and their included angle are given.