Hyperbolic trigonometry

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2020 Mathematics Subject Classification: Primary: 51M10 [MSN][ZBL]

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: \[ \frac{\sin\alpha}{\sinh a} = \frac{\sin\beta}{\sinh b} = \frac{\sin\gamma}{\sinh c} \]


[Co] H.S.M. Coxeter, "Non-Euclidean geometry", Univ. Toronto Press (1965) pp. 224–240 Zbl 0909.51003
[Co2] H.S.M. Coxeter, "Angles and arcs in the hyperbolic plane" Math. Chronicle (New Zealand), 9 (1980) pp. 17–33 Zbl 0438.51019
How to Cite This Entry:
Hyperbolic trigonometry. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article