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

146

NON-EUCLIDEAN GEOMETRY

[CH. VII.

The function cos x has here a purely analytical meaning,
being defined by the equation
x2 X4
cosx=l- + -....
2! 4!
It follows that (-'x) = cos -ki,
W(I) = oos nx,, (,X) = cos knx'
b) - co2snk
Now let x be any other value of x in the interval. If it
happens that this value is included in the set
nx1
qnx or 1
1 21A'
we know that 0 () =cos (), by the above.
But if it is not included in these forms, we can still find
positive integers m, n by going on far enough in the scale, such
that / nxl\
k 2??
where e is any positive number as small as we please.
But 0(x) and cos- are continuous functions.
It follows that = Cx
It follows that p (x) = cos ~.
This value of k will be related to the measure of the line OS,
denoted by; in the previous sections.
~ 87. We have now to deal with a rather complicated figure.
From it we shall obtain the fundamental equation of this
Trigonometry for the Right-Angled Triangle ABC, in which C
is the right angle, viz.
c a b
cos = cos cos.....................(1
Let ABC be a right-angled triangle, in which C is the right
angle.
From a point b upon AB produced draw be perpendicular
to AC.