Plane trigonometry, by S.L. Loney.

312

TRIGONOMETRY.

complex quantity is many-valued. The principal value
is that particular value of the amplitude that lies between
- 7T and + 7r.
If to the principal value of 0 we add any multiple of
27r we obtain one of its many values.
To sum up; If 0 be that value, lying between - r
and + 7r, which satisfies the equations
cos = z and sin 0 =.... (1),
VX2Y2 V y x2 + y2
then
x + y V/ = V2 + y2 [cos (2ntr + 0) + ^/-1 sin (2n7r + 0)].
The quantity 2n7r + 0 is called the amplitude and 0 is
called its principal value.
For brevity we often write equations (1) in the form
tan 0 = Y, i.e. 0 = tan-l,
X X
but it must be understood that here the angle denoted is
the one that satisfies the conditions (1).
268. De Moivre's Theorem. Whatever may be
the value of n, positive or negative, integral or fractional,
the value, or one of the values, of
(cos 0 + / -1 sin 0)n is cos nO + - 1 sin nO.
Case I. Let n be a positive integer.
By simple multiplication we have
[cos a + V - 1 sin a] [cos P + V/- 1 sin f]
= cos a cos / - sin a sin P/ + V - 1 [sin a cos /3 + cos a sin 3]
= cos (a + /3) + / -1 sin (a + /3).