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).

Proof
Since 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

return to hyperbolic trigonometry