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law of cosines

n. (context trigonometry English) A statement that relates the lengths of the side of a triangle to the cosine of one of its angle.

Law of cosines

In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states

c = a + b − 2abcosγ, 
where denotes the angle contained between sides of lengths and and opposite the side of length .

The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle is a right angle (of measure 90 °, or radians), then , and thus the law of cosines reduces to the Pythagorean theorem:

c = a + b.

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 sides of the triangle play the roles of , , and in the original formula, the following two formulas also state the law of cosines:

a = b + c − 2bccosα, 

b = a + c − 2accosβ.