Difference between revisions of "Hyperbolic trigonometry"
From Encyclopedia of Mathematics
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+ | 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: | 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 | |
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− | + | |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 | |
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− | + | |} | |
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Latest revision as of 02:04, 31 July 2012
2010 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: \[ \frac{\sin\alpha}{\sinh a} = \frac{\sin\beta}{\sinh b} = \frac{\sin\gamma}{\sinh c} \]
References
[Co] | H.S.M. Coxeter, "Non-Euclidean geometry", Univ. Toronto Press (1965) pp. 224–240 |
[Co2] | H.S.M. Coxeter, "Angles and arcs in the hyperbolic plane" Math. Chronicle (New Zealand), 9 (1980) pp. 17–33 |
How to Cite This Entry:
Hyperbolic trigonometry. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hyperbolic_trigonometry&oldid=27277
Hyperbolic trigonometry. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hyperbolic_trigonometry&oldid=27277
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article