Plane trigonometry, by S.L. Loney.

COMPLEX QUANTITIES.

311

Ex. 2. Quantity - 1 + - 3.
Here - 1+ / -1 /3 =r (cos 0 + a/ - lsin 0),
so that r cos 0 = - 1, and r sin 0 =^/3..'. r=+ 1+3= +2,
1 413
and then cos 0= and sin 0=-,
27 2
so that 0 2r.
-1+J 3=2[cos +V 1 sin
Ex. 3. Quantity - 1 /- 3.
Here r cos 0 = -1, and r sin 0= - 4/3,
1 413
so that r= + ^/1+3= + 2, cos 0= - and sin 0= -.
2 2
Hence (since we choose for 0 that value which lies between - r and
2ir
+ 7r) we have 0= -.
3
- -^/=2 [cos ( - 2)+isin ( -2)]
267. In Art. 265 the equations
cos X = and sin= -
+ ~/x2 + y2 + + /2 + y2
are satisfied by more than one value of 0. For the cosine
and sine of an angle repeat the same values when the
angle is increased by any multiple of 27r radians, so that,
if 0 denote the value between - w and + wr satisfying the
above relations, the general solution is
2nwr + 0,
where n is any integer.
This is expressed by saying that the amplitude of a