Plane trigonometry, by S.L. Loney.

318

TRIGONOMETRY.

[Exs. XLVII.]

19. Prove that
(a + bi)n + (a - bi) = 2 (a2+ b2)2 cos (-tan-).
20. If 2cos =x+-,
prove that 2 cos r =xr +
21. If 2cos0=x+-, 2cos=y+-,......
x y
prove that 2 cos (0 ++...)=xyz... +.
xyz...
22. If xr=cos +r - 1 sin 2
prove that x.. x2. 3.... ad inf. = cos 7r.
23. Using De Moivre's Theorem solve the equation
x4- x3+x2- x+ =O.
270. In Art. 269 we have only shewn that
0. 0
cos - + J- 1 sinq q
is one of the values of
(cos 0+ V -1 sin O).
The other values may be easily obtained. For
1 1
(cos 0+/- 1 sin 0)/ = [cos (2nr + 0) + J -1 sin (2n7r + 0)],
where n is any integer, and one of the values of the latter
quantity is
2n7r +. 2nr + 0
cos - + - sin.
q q
By giving n the successive values 0, 1, 2, 3,... (q- 1),
we see that each of the quantities
cos- + v - 1 sin-,
q q