The Sine Rule for Hyperbolic Triangles
For any htriangle ABC,
sin(A)/sinh(a) = sin(B)/sinh(b) = sin(C)/sinh(c).
Proof
Let α = cosh(a), β = cosh(b) and γ = cosh(c).
By appendix (1), sinh^{2}(x) = cosh^{2}(x)  1, so
(1) sinh^{2}(b) = β^{2}  1, and
(2) sinh^{2}(c) = γ^{2} 1.
The Cosine Rule then says that
sinh(b)sinh(c)cos(A) = βγ  α
Since sin^{2}(A) = 1  cos^{2}(A), this , with (1) and (2) yields
sinh^{2}(b)sinh^{2}(c)sin^{2}(A) = sinh^{2}(b)sinh^{2}(c)
 (βγ  α)^{2},
= 1  α^{2}  β^{2}  γ^{2} + 2αβγ.
Note that this expression is symmetric in α β and γ. Also note that
it is positive, so we may write it as Δ^{2} (Δ > 0).
Taking positive roots of each side of the above equation,
sinh(b)sinh(c)sin(A) = Δ
Using the symmetry of Δ, we also have
sinh(a)sinh(b)sin(C) = Δ, and
sinh(a)sinh(c)sin(B) = Δ
The theorem follows easily.

