Encyclopedia > Law of cosines

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Law of cosines

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>
iff cos C = 0 (since a, b > 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.

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