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In [[hyperbolic geometry]], the '''law of cosines''' is a pair of theorems relating the sides and angles of triangles on a [[Hyperbolic geometry|hyperbolic plane]], analogous to the planar [[law of cosines]] from plane [[trigonometry]], or the [[spherical law of cosines]] in [[spherical trigonometry]].
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Take a hyperbolic plane whose [[Gaussian curvature]] is <math>-\frac1{k^2}</math>. Then given a [[hyperbolic triangle]] ''ABC'' with angles ''&alpha;'', ''&beta;'', ''&gamma;'', and side lengths ''BC'' = ''a'', ''AC'' = ''b'', and ''AB'' = ''c'', the following two rules hold:
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:<math>\cosh\frac ak = \cosh\frac bk \cosh\frac ck - \sinh\frac bk \sinh\frac ck \cos\alpha, \,</math>
 
considering the sides, while
 
:<math>\cos\alpha = -\cos\beta \cos\gamma + \sin\beta \sin\gamma \cosh\frac ak, \,</math>
 
for the angles.
 
Christian Houzel (page 8) indicates that the hyperbolic law of cosines implies the [[angle of parallelism]] in the case of an ideal hyperbolic triangle:
:When &alpha; = 0, that is when the vertex ''A'' is rejected to infinity and the sides ''BA'' and ''CA'' are ''parallel'', the first member equals 1; let us suppose in addition that &gamma; = &pi;/2 so that cos &gamma; = 0 and sin &gamma; = 1. The angle at ''B'' takes a value &beta; given by 1 = sin &beta; cosh(''a''/k); this angle was later called ''angle of parallelism'' and Lobachevsky noted it by ''F(a)'' or &Pi;(''a'').
 
==See also==
* [[Law_of_sines#Hyperbolic_case|Hyperbolic law of sines]]
 
== External links ==
* [http://www.math.tu-berlin.de/geometrie/Lehre/SS04/VisMath/mwiki/index.php/NonEuclideanGeometry Non Euclidean Geometry, Math Wiki at TU Berlin]
 
==References==
* Houzel, Christian (1992) "The Birth of Non-Euclidean Geometry", pages 3 to 21 in ''1830&ndash;1930: A Century of Geometry'', Lecture Notes in Physics #402, [[Springer-Verlag]] ISBN 3-540-55408-4 .
* {{Cite book|last=Anderson|first=James W.|title=Hyperbolic geometry|year=2005|publisher=Springer|location=London|isbn=1-85233-934-9|edition=2nd ed.}}
* [[Miles Reid]] & Balázs Szendröi (2005) ''Geometry and Topology'', §3.10 Hyperbolic triangles and trig, [[Cambridge University Press]], ISBN 0-521-61325-6, {{MathSciNet|id=2194744}}.
* {{Cite book| last=Reiman | first=István | year=1999 | title=Geometria és határterületei | publisher=Szalay Könyvkiadó és Kereskedőház Kft. | isbn=978-963-237-012-5 }}
 
[[Category:Hyperbolic geometry]]
 
[[es:Teorema del coseno#Geometría hiperbólica]]
[[fr:Théorème d'Al-Kashi#Géométrie hyperbolique]]
[[pl:Twierdzenie cosinusów#Wzory cosinusów w geometriach nieeuklidesowych]]
 
{{geometry-stub}}

Latest revision as of 05:55, 20 October 2014

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