The Cosine Formula for Hyperbolic Triangles
If the h-triangle has a right angle at A, then
cos(B) = tanh(c)/tanh(a), and
cos(C) = tanh(b)/tanh(a).
Proof
By Pythagoras's Theorem, cosh(a) = cosh(b)cosh(c), so that
cosh(b) = cosh(a)/cosh(c).
By The Cosine Rule applied to B,
sinh(a)sinh(c)cos(B)
= cosh(a)cosh(c) - cosh(b),
= cosh(a)cosh(c) - cosh(a)/cosh(c),
= cosh(a)(cosh2(c)-1)/cosh(c),
= cosh(a)sinh2(c)/cosh(c) (by Appendix (4))
Thus, dividing by sinh(a)sinh(c), we have
cos(B)
= cosh(a)sinh(c)/sinh(a)cosh(c),
= tanh(c)/tanh(a) (by Appendix (3))
The proof for cos(C) is similar.
|
|