The Second Cosine Rule for Hyperbolic Triangles
For any h-triangle ABC,
sin(B)sin(C)cosh(a) = cos(A) + cos(B) cos(C),
with similar formulae for cosh(b) and cosh(c).
Proof
We shall use some of the notation of the proof of the Sine Rule.
We put
α = cosh(a), β = cosh(b), γ = cosh(c), and
Δ2 = 1 - α2 - β2 - γ2 + 2αβγ.
In that proof, we found that
Δ = sinh(b)sinh(c)sin(A)
= sinh(a)sinh(b)sin(C)
= sinh(a)sinh(c)sin(B)
From the Cosine Rule, we have
sinh(b)sinh(c)cos(A) = βγ - α,
sinh(a)sinh(c)cos(B) = αγ - β,
sinh(a)sinh(b)cos(C) = αβ - γ,
By appendix (1), sinh2(x) = cosh2(x) - 1, so
sinh2(a) = α2 - 1,
Then we see that, after some cancelling,
(cos(A) + cos(B)cos(C))/sin(B)sin(C)
= ((αγ - β)(βα - γ) + (βγ - α)(α2 - 1))/Δ2.
After simplification, the top line turns out to be αΔ2,
and the result follows.
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