Page 789 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

10.3 In this section we shall discuss a way to represent complex numbers using
trigonometric properties. Our work will lead to an important formula for
POLAR FORM OF A powers of complex numbers and to a method for finding nth roots of complex
COMPLEX NUMBER numbers.

Polar Form , and measures the angle from the positive real axis to the vector z, then, as

If is a nonzero complex number, (1)
suggested by Figure 10.3.1, or

so that can be written as

(2)

This is called a polar form of z.

Argument of a Complex Number

The angle is called an argument of z and is denoted by

The argument of z is not uniquely determined because we can add or subtract any multiple of from to produce another
value of the argument. However, there is only one value of the argument in radians that satisfies

This is called the principal argument of z and is denoted by

Figure 10.3.1

EXAMPLE 1 Polar Forms
Express the following complex numbers in polar form using their principal arguments:

(a)