Hyperbolic law of cosines

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 − 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 ⁡ a k = cosh ⁡ b k cosh ⁡ c k − sinh ⁡ b k sinh ⁡ c k cos ⁡ α ,

considering the sides, while

cos ⁡ α = − cos ⁡ β cos ⁡ γ + sin ⁡ β sin ⁡ γ cosh ⁡ 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).