View source for Hyperbolic trigonometry

{{MSC|51M10}}
{{TEX|done}}

The trigonometry on the Lobachevskii plane (cf. [[Lobachevskii   geometry]]). Let $a$, $b$, $c$ be the lengths of the sides of a triangle on the Lobachevskii plane, and let $\alpha$, $\beta$, $\gamma$ be the angles of this triangle. The following relationship (the cosine theorem), which relates these sides and angles, is valid:
\[
\cosh a = \cosh b \cosh c - \sinh b \sinh c \cos \alpha.
\]
All the remaining relations of hyperbolic trigonometry follow from this one, such as the so-called sine theorem:
\[
\frac{\sin\alpha}{\sinh a} = 
\frac{\sin\beta}{\sinh b} =
\frac{\sin\gamma}{\sinh c}
\]

====References==== 
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|valign="top"|{{Ref|Co}}||valign="top"| H.S.M. Coxeter, "Non-Euclidean geometry", Univ. Toronto Press (1965) pp. 224–240 {{ZBL|0909.51003}}
|-
|valign="top"|{{Ref|Co2}}||valign="top"| H.S.M. Coxeter, "Angles and arcs in the hyperbolic plane" ''Math. Chronicle (New Zealand)'', '''9''' (1980) pp. 17–33
|-
|}