Complex Numbers, Polar Equations, and Parametric Equations · 2017-05-11 · Example 4 SOLVING AN EQUATION (COMPLEX ROOTS) Find all complex number solutions of x5 –i = 0. Graph

Copyright © 2017, 2013, 2009 Pearson Education, Inc. 1

8Complex

Numbers,

Polar

Equations, and

Parametric

Equations

Copyright © 2017, 2013, 2009 Pearson Education, Inc.


De Moivre’s Theorem; Powers

and Roots of Complex Numbers8.6

Powers of Complex Numbers (De Moivre’s Theorem) ▪ Roots of Complex Numbers

Copyright © 2017, 2013, 2009 Pearson Education, Inc. 38.4-3

De Moivre’s Theorem

is a complex number, and if n is any real number, then

In compact form, this is written


Example 1 FINDING A POWER OF A COMPLEX

NUMBER

Find and express the result in rectangular

form.

First write in trigonometric form as

2(cos60 sin60 ).i


Example 1 FINDING A POWER OF A COMPLEX

NUMBER (continued)

Now apply De Moivre’s theorem.

480° and 120° are coterminal.

Rectangular form


nth Root

For a positive integer n, the complex number a + bi is an nth root of the complex number x + yi if


nth Root Theorem

If n is any positive integer, r is a positive real number, and θ is in degrees, then the nonzero complex number r(cos θ + i sin θ) has exactly n distinct nth roots, given by

where


Note

In the statement of the nth root theorem,

if θ is in radians, then for 0, 1, 2, ..., 1k n


Example 2 FINDING COMPLEX ROOTS

Find the two square roots of 4i. Write the roots in

rectangular form.

Write 4i in trigonometric form:

The square roots have absolute value

and argument


Example 2 FINDING COMPLEX ROOTS (continued)

Since there are two square roots, let k = 0 and 1.

Using these values for , the square roots are




Example 3 FINDING COMPLEX ROOTS

Find all fourth roots of Write the roots in

rectangular form.

Write in trigonometric form:

The fourth roots have absolute value

and arguments



Since there are four roots, let k = 0, 1, 2, and 3.

Using these values for α, the fourth roots are

2 cis 30°, 2 cis 120°, 2 cis 210°, and 2 cis 300°.





The graphs of these roots lie on a circle with center at the origin and radius 2. The roots are equally spaced about the circle, 90° apart.


Example 4 SOLVING AN EQUATION (COMPLEX

ROOTS)

Find all complex number solutions of x5 – i = 0. Graph

them as vectors in the complex plane.

There is only one real solution, 1, but there are five

complex number solutions.

Write 1 in trigonometric form:


Example 4 SOLVING AN EQUATION (COMPLEX

ROOTS) (continued)

The fifth roots have absolute value and

arguments

Since there are five roots, let k = 0, 1, 2, 3, and 4.

Solution set: {cis 0°, cis 72°, cis 144°, cis 216°, cis 288°}


SOLVING AN EQUATION (COMPLEX

ROOTS) (continued)

The tips of the arrows representing the five fifth roots all lie on a unit circle and are equally spaced around it every 72°, as shown.

Example 4

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Complex Numbers, Polar Equations, and Parametric Equations · 2017-05-11 · Example 4 SOLVING AN EQUATION (COMPLEX ROOTS) Find all complex number solutions of x5 –i = 0. Graph