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

104 NON-EUCLIDEAN GEOMETRY

[CH. IV.

Therefore
sinh a: sinh b: sinh c = sech 1: sech m: sech n.
If one of the angles is obtuse, we obtain the same result,
using the notation II ( - x) = r - (x).
For the right-angled triangle, the result follows from ~ 59, I.
II. We shall now prove the theorem corresponding to the
Cosine Rule of ordinary Trigonometry.
We take in the first place the case when B and C are acute
angles.
From A draw the perpendicular AD to BC.
Let AD=j?, CD=q, and BD=a-q (Fig. 75).
Then, from the triangle ABD we have
cosh c = cosh(a - q) coshp (~ 59, IV.),
and from the triangle ACD we have
cosh b = coshp cosh q.
Also, we have
tanh(a - q) = tanh c tanh m (~ 59, VI.).
cosh c cosh q
Therefore cosh b osh c osh q
cosh(a - q)
= cosh c (cosh a cosh (a - q) - sinh a sinh (a - q))
cosh (a- q)
= cosh a cosh c - sinh a cosh c tanh (a - q)
= cosh a cosh c - sinh a sinh c tanh m.
If the angle B is obtuse, so that D falls on CB produced,
the same result follows, provided account is taken of the
notation - ()
If the angle B is a right angle, the result follows from
~59, IV.
We are thus brought to the Cosine Formula, which may
be put in the form:
cosh a = cosh b cosh c - sinh b sinh c tanh 1.
~61. The Measurement of Angles.
Up till this stage, except in ~~ 51-2, there has been no
need to introduce a unit of angle into our work. The