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.