Plane trigonometry, by S.L. Loney.

314 TRIGONOMETRY.
Case III. Let n be fractional and equal to -, where
q
q is a positive integer and p is an integer, positive or
negative.
By the previous cases, we have
cos - + V/ -1 sin - = cos (q. ) +/ -1 sin (q.
q q- q
= cos 0 + /- 1 sin 0.
Therefore cos - + /-1 sin - is such that when multiq q
plied by itself q times it gives cos 0 + V - 1 sin 0.
Hence cos -+ V - 1 sin - is one of the qth roots of
q q
cos 0 + V - 1 sin 0,
0.0
i.e. cos- + / -1 sinq q
is one of the values of
1
(cos 0 + J-1 sin 0).
Raise each of these quantities to the pth power.
We then have that one of the values of
[cos 0 + /-1 sin 0]q is (cos - + V /-1 sin -
q /
i.e. is cos - + V/-1 sin -
q q
269. The quantity i is always used to denote - 1
and will be often so used hereafter. The expression
cos 0 + i sin 0 therefore means cos 0 + / - 1 sin 0.