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

212 INTRODUCTION TO MATHEMATICS
This property (A) is an example of what
is called an addition-theorem. When any
function [say f(x)] has been defined, the first
thing we do is to try to express f(x+y) in
terms of known functions of x only, and
known functions of y only. If we can do so,
the result is called an addition-theorem.
Addition-theorems play a great part in
mathematical analysis. Thus the additiontheorem for the sine is given by
sin (x+y) =sin x cos y+cos x sin y,
and for the cosine by
cos (x+y) =cos x cos y-sin x sin y.
As a matter of fact the best ways of defining sin x and cos x are not by the elaborate
geometrical methods of the previous chapter,
but as the limits respectively of the series
X3 X5 X7
- î+ - -t-7+ etc...,
3! 5! 7!
x2 X4 x6
and 1- + - + etc.....,
a 2! 4! 6!
so that we put
x3 x5 x7
sin x =x - + - 7+ etc....,
X2 X4 X6
cos x=1- +4- + - etc...,
92! 4! 6!