# hyperbolic trigonometry

 The Cosine Rule for Hyperbolic Triangles For any h-triangle ABC, sinh(b)sinh(c)cos(A) = cosh(b)cosh(c) - cosh(a), with similar formulae for cos(B) and cos(C). ProofSince angles and lengths are invariant under H(2), we may as well assume that A is O, and that B is on the positive real axis. As A = O, AB and AC are line segments, making the angle A. Thus the complex coordinates of A, B and C are 0, r and seiA, with 0 < r,s < 1. The hyperbolic lengths of the sides are given by (1) c = d(0,r) = 2arctanh(r), so r = tanh(c/2), (2) b = d(0, seiA) = 2arctanh(s), so s = tanh(b/2), and (3) a = d(seiA, r) = 2arctanh(|seiA - r|/|rseiA -1|). From (3), tanh2(a/2) = |seiA -r|2/|rseiA-1|2                = (s2 + r2 - 2rscos(A)/(r2s2 + 1- 2rscos(A)) Now, by appendix (5), cosh(a) = (1 + tanh2(a/2))/(1 -1 tanh2(a/2)). After a bit of algebra, (1),(2) and appendix (4),(5), cosh(a) = (r2s2 + r2 + s2 + 1 -4rscos(A) /(r2s2 -r 2 - s2 +1).            = ((1 + s2)(1 + r2) - 4rscos(A))/(1 - s2)(1 - r2)).            = cosh(b)cosh(c) - sinh(b)sinh(c)cos(A). This immediately gives the required result. by the Basic Strategy