The Law of Cosines
The law of cosines, a² = b² + c² − 2bc cos A, extends Pythagoras to every triangle by adding a correction term built from the included angle.
--
The law of cosines, a² = b² + c² − 2bc cos A, extends Pythagoras to every triangle by adding a correction term built from the included angle.
Pythagoras' theorem only speaks about right triangles. The law of cosines extends it to every triangle by adding one correction term built from the included angle.
Each side has its own version — the pattern is always "square of the unknown side = sum of the squares of the other two, minus twice their product times the cosine of the angle between them":
At we have , the extra term vanishes, and the law becomes — Pythagoras' theorem.
The term also explains how the third side reacts: a small angle gives a large positive cosine, so more is subtracted and comes out short. An obtuse angle makes the cosine negative, so the term is added and comes out long.
In triangle ABC: , and . Find .
| Unknown | Formula | Given |
|---|---|---|
| Side a | a² = b² + c² − 2bc cos A | SAS |
| Side b | b² = a² + c² − 2ac cos B | SAS |
| Side c | c² = a² + b² − 2ab cos C | SAS |
| An angle | cos A = (b² + c² − a²) ÷ 2bc | SSS |
SAS — two sides and the angle between them known: find the third side. SSS — all three sides known: rearrange to find any angle.