Let z" = r" [cos (0 + 2 kπ) + j sin (0 + 2 kπ)] be the number.

Let

COS

k = n, then z1 = r
= r [cos (2 + 2) + j sin (+2n)]
=r[cos (+2)+j sin(+2)].

COS

But zo 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.

The roots 20, 21, Zn-1, all have their terminal points on the circumference of a circle of radius r. Their amplitudes differ by

Then

and

tan-1 = 68° 11.9'.

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

10

z = 29 [cos (13° 38.4' + k) +j sin (13° 38.4' + 3 km)].

2=

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 ne and cos no.-By DeMoivre's Theorem and the binomial theorem, cos no + j sin no (cos +j sin 0)" = cos" + nj cos"-10 · sin 0

+

n (n − 1)
COS"
12

n-20 sin2 0

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

and (3) may be used to compute the functions of angles. Thus,

where a, b, c, . . . may be computed as follows:

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

(a) sin 20°.

(b) cos 25°.

(c) tan 30°.
(d) sin 45°.

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)

x2

e2 = 1 − x +

+ + +

But, by Art. 99, the expressions in the first and second parentheses are equal to cos 0 and sine respectively.

(2)

(3)

.. eio = cos 0+ i sin 0.

Substituting x = — i0 in (1) and reducing as before, we have

Note. - The expressions for sin e, cose, and tan 6, given in (4), (5), and (6), are called exponential values of these functions. They are also called Euler's Equations after Euler their discoverer. Euler (1707-1783) was one of the greatest Physicists, astronomers and mathematicians of the eighteenth century.

EXERCISES

By means of the exponential values prove the following identities: