The law of cosines is a statement about arbitrary triangles in the plane which generalizes the Pythagorean theorem by correcting it with a term proportional to the cosine of the opposing angle. Let a, b, c be the sides of the triangle and A, B, C the angles opposite those sides. Then
- <math>c^2 = a^2 + b^2 - 2ab \cos C </math>
This formula is useful to compute the third side of a triangle when two sides and the enclosed angle are known, and to compute the angles of a triangle if all three sides are known.
The law of cosines also shows that
- <math>c^2 = a^2 + b^2</math>
= 0 (since a
> 0), which is equivalent to C
being a right angle. (In other words, this is the Pythagorean Theorem and its converse.)
Also see triangulation, law of sines and trigonometry.