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

100

NON-EUCLIDEAN GEOMETRY

[CH. IV.

~ 59. The Equations connecting the Sides and Angles
of a Right-Angled Triangle.
Let ABC be any right-angled triangle, C being the right
angle.
Produce the sideAC through C, and draw the parallel Bt2
through B to AC.
A I
B
s n
2
L
FIG. 73.
Also produce AB through B to L, where AL is the segment 1,
such that =II (). [X=L BAC (cf. Fig. 37).]
Through L draw LQ2 parallel to BQI and AC.
Let the Limiting-Curves through B and L, with centre at
2, meet the axes at B', D, and D' (Fig. 73).
Let the arcs BB', DD', LD be denoted by s, s1, s2, and let
the segment BD =u.
Then we have
Ssinha=s=seu, [~ 57 (2).]
s +s2 = Stanh, [~ 56(4).]
s2 = S tanh BL = S tanh (I - c),
eu= cosh BL=cosh (- c). [~ 56 (3).]
It follows that
sinh a = cosh (I - c){tanh I - tanh (I - c)}
sinh I cosh (I - c) - cosh 1 sinh (I - c)
cosh I

= sinh c/cosh 1.