The hyperbolic law of cosines is an identity in hyperbolic geometry relating the measure of an angle of a triangle in the hyperbolic plane with the lengths of its three sides.
In particular, for the figure at right with vertices (and angle measures) \(A\), \(B\), and \(C\) and opposite side lengths \(a\), \(b\), and \(c\), the hyperbolic law of cosines is—
\[\cosh c = \cosh a \cosh b - \sinh a \sinh b \cos C\]
There is also a dual law relating a single side length with the three angle measures:
\[\cos C = -\cos A \cos B + \sin A \sin B \cosh c\]
There are also analogous identities for more general types of triangles in the hyperbolic plane. We discuss these in the next section.
Laws for generalized hyperbolic triangles
TODO
