Law of cosines

Statement

Let $ABC$ be a triangle in the Euclidean space $\mathbb{R}^n$. Let $a = d(B, C)$, $b = d(C, A)$ and $c = d(A, B)$ be its side lengths, where $d$ is the Euclidean distance. Let $\gamma = \angle ACB$ be its angle at $C$. Then $$c^2 = a^2 + b^2 - 2ab \cos \gamma.$$

Remarks

The proof in vector form is short. With $u = A - C$ and $v = B - C$: $c^2 = \|u - v\|^2 = \|u\|^2 + \|v\|^2 - 2 u \cdot v$, and $u \cdot v = ab \cos \gamma$. For $\gamma = \pi / 2$ the formula is the Pythagorean theorem. The formula lets you find the third side from two sides and the angle between them. It also gives each angle from the three sides. Mathlib states the formula for all points, with its convention for the angle at a repeated point.

History

Book II of Euclid's Elements contains the statement in the language of areas of rectangles. Propositions 12 and 13 of Book II treat the obtuse and the acute case.

Explanations

L1 human

The law of cosines corrects the Pythagorean theorem for triangles without a right angle. If the angle $\gamma$ is larger than a right angle, $\cos \gamma < 0$, and the opposite side is longer than in the right triangle. If $\gamma$ is smaller, the opposite side is shorter.