An introduction to mathematics, by A. N. Whitehead.

TRIGONOMETRY 189
see that, if u be chosen to have any value, the
arguments u and 2Tr +u, and 47r+u, and
6r +u, and 87r+u and so on indefinitely,
have all the same values for the corresponding sines and cosines. In other words,
sin u = sin (2Tr +u) =sin (47r+u) =sin (67r +u)
=etc.;
cos u = cos (2wr +u) = cos (47r +u) = cos (67r +u)
= etc.
This fact is expressed by saying that sin u
and cos u are periodic functions with their
period equal to 27r.
The graph of the function y = sin x (notice
that we now abandon v and u for the more
familiar y and x) is shown in fig. 28. We
take on the axis of x any arbitrary length at
pleasure to represent the number 7r, and on
the axis of y any arbitrary length at pleasure
to represent the number 1. The numerical
values of the sine and cosine can never exceed unity. The recurrence of the figure
after periods of 27r will be noticed. This
graph represents the simplest style of periodic
function, out of which all others are constructed. The cosine gives nothing fundamentally different from the sine. For it is
easy to prove that cos x = sin (x+ 7); hence
it can be seen that the graph of cos x is simply
fig. 28 modified by drawing the axis of OY