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Law of cosines

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\begin{matrix} c^{2} & = a^{2} + b^{2} - 2 a b \cos γ \\ a^{2} & = b^{2} + c^{2} - 2 b c \cos α \\ b^{2} & = a^{2} + c^{2} - 2 a c \cos β \end{matrix}

the law of cosines generalises the pythagorean theorem to all triangles.

The law of cosines can be used to solve triangles:

the third side of a triangle two sides and their shared angle is known:
$c = \sqrt{a^{2} + b^{2} - 2 a b \cos γ}$
the angles of a triangle if the three sides are known:
$γ = \arccos (\frac{a^{2} + b^{2} - c^{2}}{2 a b})$
the third side of a triangle if two sides and an angle opposite to one of them is known:
$a = b \cos γ \pm \sqrt{c^{2} - b^{2} \sin^{2} γ}$

Law of cosines

See also