Plane trigonometry, by S.L. Loney.

366 TRIGONOMETRY.
Circular functions of complex angles.
307. When x is a complex quantity, the functions
sin x and cos x have at present no meaning.
For real values of x we have already shewn in Arts.
279 and 280 that
Q3 '5 X7
sin x=x +.....ad inf.!3 1.5 7
X2 X4 X6
and cos x = 1 — 6 +-....ad inf.
Let us define sin x and cos x, when x is complex,
so that these relations may always be true, i.e. for all
values of x let
X3 X5 X7
sinx = -- + x +(1),
sI- + [5 I7 +...............(1),
X2 X4 X6
and cosx= 1- +- — +............... (2).
When x is complex, the quantities sin x and cos x are
then only short ways of writing the series on the right-hand
sides of (1) and (2).
308. We have then, for all values of x, real or
complex,
X2 X3i X4
cosx + sinx=l+ - - +......
(Xi)2 (Xi)3 (Xi)4
= 1+ xi + -- + +..
=- exi (Art. 302.)

So

\.....

cos x - i sin x = e-x.