# Law of cosines

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This article is about the law of cosines in Euclidean geometry. For the cosine law of optics, see Lambert’s cosine law.

In trigonometry, the

**law of cosines**(also known as the**cosine formula**or**cosine rule**) is a statement about a general triangle that relates the lengths of its sides to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states thatwhere γ denotes the angle contained between sides of lengths

*a*and*b*and opposite the side of length*c*.The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle

*γ*is a right angle (of measure 90° or π/2 radians), then cos(γ) = 0, and thus the law of cosines reduces toThe law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known.

By changing which legs of the triangle play the roles of

*a*,*b*, and*c*in the original formula, one discovers that the following two formulas also state the law of cosines: