These three roots may be represented graphically as follows: Draw a circle of radius 1, Fig. 112. Since the moduli of 20, 21, and 22 are each 1, their terminal points will lie on the circumference of the circle. The amplitudes of 20, 21, and 22 are 0, 3 π, and respectively. Therefore 20 = Example. Find the three cube roots of 23

OA, 21

OB, and 22

=

-8.

OC.

FIG. 112.

=

Solution. Plot -8 OA, Fig. 113. The amplitude is π and the modulus is 8.

Then

Let k = 2, then 22 = 2 [cos § π + j sin § π] = 1 − j √3.

Now the moduli of 20, 21, and 22 are each equal to 2. The points representing the cube roots are therefore on the circumference. of a circle of radius 2, Fig. 113. Constructing the amplitudes 60°, 180°, and 300°, we have zo = OB, 21 = OC, and 22 OD. 98. To find the nth roots of a number. · An nth root of a number is one of its n equal factors. By DeMoivre's Theorem, when a number is raised to the nth power, the modulus is raised to the nth power, while the amplitude is multiplied by n. Then to extract the nth root, we extract the positive nth root of the modulus, and divide the amplitude by n.*

* The student is already familiar with the fact that there is one positive square root of a positive number, and one positive cube root of a positive number. It is true, in general, that there is one and only one positive nth root of any positive number.

Let zn = pn [cos (0 + 2 kπ) + j sin (0 + 2 kπ)] be the number.

Let

k

4

A = 2, then a = roos (+)+ sin(+1)]

22

[

COS

n

But z。 and 2 are seen to be the same, having the same moduli, and amplitudes differing by 360°. Any other integral value of k will lead to one of the n values of z already found as may be easily shown. There are therefore but n nth roots of a number.

Then

and1

25 = √29 [cos (68° 11.9′ + 2 kπ) +j sin (68° 11.9' + 2 kπ)],

2 =

10

29 [cos (13° 38.4′ + ‡ kπ) + j sin (13° 38.4′ + ‡ kπ)].

Assigning to k the values 0, 1, 2, 3, 4, we have

10

In Fig. 114, OA represents the original vector 2 +5j. Construct a circle with radius 29 1.4 approximately. Lay off OB with the amplitude 13° 38.4′, and OC, OD, OE, and OF respectively every 72° from OB. Then OB 20, OC 21, OD = 22, OE 23, and OF = 24.

=

In the following exercises, find and plot all the values of z.

99. Expansion of sin no and cos no.-By DeMoivre's Theorem and the binomial theorem, cos no + j sin no = (cos +j sin 0)"= cos" 0 + nj cos"-1 0 · sin 6

while n and are to vary. Substituting these values,

Equating the coefficients of the imaginary parts of (1),

100. Computation of trigonometric functions. Formulas (2) and (3) may be used to compute the functions of angles. Thus,

From the table of natural functions sin 10°

By means of (2), cos 10° may be computed.

0.17364.

0.17365.

EXERCISES

Compute the following functions correct to the fourth decimal place and compare with the tables.

101. Exponential values of sin e, cos 0, and tan 0.— In algebra it is proved that if e is the base of the natural system of logarithms, then

(1)

ех

=

1 + x +

++++..