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

59, 60] THE OBLIQUE-ANGLED TRIANGLE

103

~ 60. The Equations for an Oblique-Angled Triangle.
In the case of the Oblique-Angled Triangle ABC, the sides
opposite the angular points A, B, and C will be denoted by a, b,
and c, as usual; but the angles at A, B, and C will be denoted
by X, t1, and v.
With this notation the distance of parallelism for the angle
at A will be 1.
We proceed to prove that
I. sinh a: sinh b: sinh c = sech 1: sech m: sech n.
This corresponds to the Sine Rule of ordinary Trigonometry.
A
C 5 \
B a-q O q C
FIG. 75.
Let ABC be all acute angles.
From an angular point, say A, draw the perpendicular AD to
the opposite side. We then obtain two right-angled triangles
ABD and ACD, as in Fig. 75.
Writing AD =p, we have (by ~ 59, I.)
sinh c
sinhp = cosh, from the triangle ABD,
sinh b
and sinhp= csh, from the triangle ACD.
s cosh '
Thus we have
sinh b: sinh c = sech,n: sech n.
Taking another angular point-say B-and proceeding in
the same way, we would have

sinh a: sinh c = sech 1: sech n.