The Sine Rule for Hyperbolic Triangles
For any h-triangle 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), sinh2(x) = cosh2(x) - 1, so
(1) sinh2(b) = β2 - 1, and
(2) sinh2(c) = γ2 -1.
The Cosine Rule then says that
sinh(b)sinh(c)cos(A) = βγ - α
Since sin2(A) = 1 - cos2(A), this , with (1) and (2) yields
sinh2(b)sinh2(c)sin2(A) = sinh2(b)sinh2(c)
- (βγ - α)2,
= 1 - α2 - β2 - γ2 + 2αβγ.
Note that this expression is symmetric in α β and γ. Also note that
it is positive, so we may write it as Δ2 (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.
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