TheoremProved

Law of cosines

Statement

In any triangle with side lengths $a$ , $b$ , $c$ and interior angle $\gamma$ opposite the side of length $c$ , $c^2 = a^2 + b^2 - 2ab\cos\gamma$ .

Why is it true?

The law of cosines is the Pythagorean theorem with a correction term $-2ab\cos\gamma$ that accounts for how far the angle $\gamma$ departs from a right angle. When $\gamma = 90^\circ$ , $\cos\gamma = 0$ and the formula reduces to $c^2 = a^2 + b^2$ ; opening $\gamma$ into an obtuse angle makes $\cos\gamma < 0$ so $c^2$ exceeds $a^2 + b^2$ , while closing $\gamma$ into an acute angle makes $\cos\gamma > 0$ and pulls the opposite side $c$ shorter.

Proof sketch

Drop the altitude $h$ from the vertex opposite side $b$ onto the line containing side $b$ . In terms of the angle $\gamma$ , this altitude has length $h = a\sin\gamma$ and meets the line of side $b$ at signed distance $a\cos\gamma$ from the vertex of $\gamma$ , leaving a segment of length $|b - a\cos\gamma|$ to the other endpoint. Applying the Pythagorean theorem to the right triangle with legs $h$ and $|b - a\cos\gamma|$ and hypotenuse $c$ gives $c^2 = (a\sin\gamma)^2 + (b - a\cos\gamma)^2 = a^2(\sin^2\gamma + \cos^2\gamma) + b^2 - 2ab\cos\gamma = a^2 + b^2 - 2ab\cos\gamma$ .

Proved by

proved -300Euclid of Alexandria

Topics that use this theorem

Trigonometric relations in triangles Trigonometry

Related theorems

Pythagorean theorem Law of sines Heron's formula

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Euclid (translated by Thomas L. Heath) (1956). The Thirteen Books of Euclid's Elements, Vol. 1 (Book II, Propositions 12–13)
Glen Van Brummelen (2009). The Mathematics of the Heavens and the Earth: The Early History of Trigonometry