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

72, 73] TiE AREA OF A TRIANGLE 125
Let P be any point on OB, and PM the perpendicular from
P to QA.
Then, from the associated right-angled triangle for the
quadrilateral OMPC, we have
m
tanh Y cosh -cosh (~59, V.)
l lo
But the area of the quadrilateral OABO is given by
J J cosh dx dy.
Denote this by S.
Integrating, we have
S=k sinh dx
dx
a cosh- dx
k dx~~
m x
-kTasin2csinh2 '-; inh2
sinh m
=kk2ssn-1
sisih
sinh m
S ik
sinp-sn
But, from the associated right-angled triangle, we have
sinha
tanh b = k (~59, 111.)
sinh k
b
And tanh =cos/3. (~62.)