1. Copyright © 2017, 2013, 2009 Pearson Education, Inc.8
Complex Numbers, Polar Equations, and
Parametric Equations
Copyright © 2017, 2013, 2009 Pearson Education, Inc. 1

2. De Moivre’s Theorem; Powers and Roots of Complex Numbers
8.6
De Moivre’s Theorem; Powers and Roots of Complex Numbers
Powers of Complex Numbers (De Moivre’s Theorem) ▪ Roots of Complex Numbers

3. De Moivre’s Theorem
is a complex number, and if n is any real number, then
In compact form, this is written

4. Find and express the result in rectangular form.
Example 1
FINDING A POWER OF A COMPLEX NUMBER
Find and express the result in rectangular form.
First write in trigonometric form as

5. Now apply De Moivre’s theorem.
Example 1
FINDING A POWER OF A COMPLEX NUMBER (continued)
Now apply De Moivre’s theorem.
480° and 120° are coterminal.
Rectangular form

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

7. 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

8. Note
In the statement of the nth root theorem, if θ is in radians, then

9. Find the two square roots of 4i. Write the roots in rectangular form.
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

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

11. Example 2
FINDING COMPLEX ROOTS (continued)

12. Find all fourth roots of Write the roots in rectangular form.
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

13. Since there are four roots, let k = 0, 1, 2, and 3.
Example 3
FINDING COMPLEX ROOTS (continued)
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°.

14. Example 3
FINDING COMPLEX ROOTS (continued)

15. Example 3
FINDING COMPLEX ROOTS (continued)
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.

16. Write 1 in trigonometric form:
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:

17. The fifth roots have absolute value and arguments
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°}

18. Example 4
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.