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.
Figure 1 – A triangle. The angles α,β, and γ are respectively opposite the sides a, b, and c.
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 that
where 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 to
The 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:
