Alternating polynomial

In hyperbolic geometry, the law of cosines is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry.

Take a hyperbolic plane whose Gaussian curvature is $-\frac{1}{k^2}$. Then given a hyperbolic triangle ABC with angles α, β, γ, and side lengths BC = a, AC = b, and AB = c, the following two rules hold:

$\cosh \frac{a}{k}=\cosh \frac{b}{k}\cosh \frac{c}{k}-\sinh \frac{b}{k}\sinh \frac{c}{k}\cos \alpha ,$

considering the sides, while

$\cos \alpha =-\cos \beta \cos \gamma +\sin \beta \sin \gamma \cosh \frac{a}{k},$

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 α = 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 γ = π/2 so that cos γ = 0 and sin γ = 1. The angle at B takes a value β given by 1 = sin β cosh(a/k); this angle was later called angle of parallelism and Lobachevsky noted it by F(a) or Π(a).

See also

• Hyperbolic law of sines

External links

• Non Euclidean Geometry, Math Wiki at TU Berlin

References

• Houzel, Christian (1992) "The Birth of Non-Euclidean Geometry", pages 3 to 21 in 1830–1930: A Century of Geometry, Lecture Notes in Physics #402, Springer-Verlag ISBN 3-540-55408-4 .
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• Miles Reid & Balázs Szendröi (2005) Geometry and Topology, §3.10 Hyperbolic triangles and trig, Cambridge University Press, ISBN 0-521-61325-6, Template:MathSciNet.
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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