Law of Cosines

The law of cosines is the Pythagorean theorem with a correction term: when angle C is exactly 90°, cos C vanishes and c² = a² + b² reappears. Open the angle wider and the correction adds length; squeeze it and the correction subtracts. Euclid proved both cases geometrically around 300 BCE, and the Persian astronomer al-Kashi gave the modern trigonometric form in the 1400s: in France the result still carries his name. A surveyor's example: from one station, two landmarks lie 8 km and 5 km away with 60° between the sightlines, so their separation is $\sqrt{64 + 25 - 2 \cdot 8 \cdot 5 \cdot \cos 60^\circ} = \sqrt{49} = 7$ km.

Solved for the angle, C = arccos((a² + b² − c²)/(2ab)) uses the principal branch of arccos, spanning 0° to 180°: precisely the range of a triangle's interior angle, so the answer is unique with no ambiguous case. The three sides must obey the triangle inequality, or the arccos argument escapes [−1, 1] and no triangle exists.