The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

62, 63, 64, 65] TRIGONOMETRICAL FORMULAE 109
If wve insert these values in the Trigonometrical Formulae
of ~ 59, we obtain:
sinh a = sinh c sin X from sinh c =sinh a cosh 1.
sinh b = tanh acot,,, sinh b =tanh asinhlI.
coshec = cot Xcotx,, coshe =sinhlIsinh m.
cosh c = cosh acosh b,,cosh c =cosh acosh b..
cos X = cosh asin,~,, cosh a tanhlIcosh m.
tanh a = tanh c cos u,,tanh a tanh m, tanh c.
And the formulae of ~ 60 f or the Oblique-Angled Triangle
beoesin ha: sin~h b: sinh c= sin X: sinMA: sin v,
cosh a = cosh b cosh- c - sinh b sinh c cos X
All1 these results agree with the corresponding formulae in
S~pherical Trigonometry, when X, At, v take the Place of A, B, C, and
the Hyperbolic Functions of a, b, and c take the place of the circular
Functions of a, b, and c.
~ 64. The Angle of Parallelism.
Since tanh a cosoL-,
we have I1-coscx. 1 -tanha
1 +cosou. 1+tanha'
Theref ore tan.2 e -2
and tan ~e -a.
The angle u. is acute, so the positive sign has to be taken
in extracting the square root.
This may be written
tan IIIl(p)=e-eP.*
~ 65. The formulae of ~~ 56-64 have been deduced on the
understanding that the unit of length employed is the distance
between concentric Limiting-Curves when the ratio of the arcs
cut off by two of their axes is e.
*This result is given by JBolyai, Appendix, ~ 29, and by Lobatschewsky
in his various books, e.g. Geometrische Untersuchungen zur Theorie, der
Peeralteltinien, ~ 36.