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Difference between revisions of "Hyperbolic trigonometry"

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The trigonometry on the Lobachevskii plane (cf. [[Lobachevskii geometry|Lobachevskii geometry]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048350/h0483501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048350/h0483502.png" />, be the lengths of the sides of a triangle on the Lobachevskii plane, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048350/h0483503.png" /> be the angles of this triangle. The following relationship (the cosine theorem), which relates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048350/h0483504.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048350/h0483505.png" />, is valid:
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{{MSC|51M10}}
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{{TEX|done}}
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048350/h0483506.png" /></td> </tr></table>
 
  
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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:
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\[
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\cosh a = \cosh b \cosh c - \sinh b \sinh c \cos \alpha.
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\]
 
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:
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\[
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\frac{\sin\alpha}{\sinh a} =
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\frac{\sin\beta}{\sinh b} =
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\frac{\sin\gamma}{\sinh c}
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\]
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048350/h0483507.png" /></td> </tr></table>
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====References====  
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{|
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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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====Comments====
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|-
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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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====References====
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|}
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.S.M. Coxeter,   "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 224–240</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,   "Angles and arcs in the hyperbolic plane" ''Math. Chronicle (New Zealand)'' , '''9''' (1980) pp. 17–33</TD></TR></table>
 

Revision as of 00:04, 31 July 2012

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

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://encyclopediaofmath.org/index.php?title=Hyperbolic_trigonometry&oldid=19080
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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