Plane trigonometry, by S.L. Loney.

FACTORS OF Xn + 1.

431

Hence we have
r2l/ 2r7r
x-l=(x2- 1) II (-2x cos -—. 1,
r=l -1
when n is even, and
n-i
x -1=(x-1) n (2-2xcos +1)
when n is odd.
These formulae can also be deduced from the fundamental one of Art. 362 by putting nO = 27r.
367. To resolve x + 1 into factors.
We must solve the equation
n + 1 =0,
i.e. n = - 1 = cos (2r7r + 7r) + i sin (2r7r + 7r),
where r is any integer,
so that x = {cos (2r7r + 7r) + i sin (2r7r + 7r)}n
2r7r + 7r. 2r7r + 7r
= cos — + i sin...........
n n
First, let n be even.
As in Art. 271, the values of the expression (1) are
7r.. 7 37.. 37 57r.. 57r
cos - + i sin- cos - i sin-, cos - + sinn - n n - n n n
(n-1).. (n - 1)7r... cos + i sin
n n
The factors corresponding to the first of these pairs are
7r.. r r..
- cos - -isin and x - cos - + i sin -,
n n n n
i.e. the quadratic factor
'7/
22- 2x cos -+ 1.
n