This document provides an overview of De Moivre's theorem and nth roots of complex numbers. It includes definitions of key concepts like the complex plane, trigonometric form of complex numbers, multiplication and division of complex numbers, powers of complex numbers, and roots of complex numbers. Examples are provided to demonstrate finding trigonometric forms, multiplying complex numbers, using De Moivre's theorem, and finding cube roots. The document is intended to teach readers how to extend equation solving techniques to include equations of the form zn = c, where n is an integer and c is a complex number.

1. 6.6
De Moivre’s
Theorem and
nth Roots
Copyright © 2011 Pearson, Inc.

2. What you’ll learn about
 The Complex Plane
 Trigonometric Form of Complex Numbers
 Multiplication and Division of Complex Numbers
 Powers of Complex Numbers
 Roots of Complex Numbers
… and why
The material extends your equation-solving technique
to include equations of the form zn = c, n is an integer
and c is a complex number.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 2

3. Complex Plane
Copyright © 2011 Pearson, Inc. Slide 6.1 - 3

4. Absolute Value (Modulus) of a
Complex Number
The absolute value or modulus
of a complex number
z  a  bi z  a  bi  a 
b
is | | | | .
2 2
a bi a bi
In the complex plane, | | is the distance of
from the origin.
 
Copyright © 2011 Pearson, Inc. Slide 6.1 - 4

5. Graph of z = a + bi
Copyright © 2011 Pearson, Inc. Slide 6.1 - 5

6. Trigonometric Form of a Complex
Number
The trigonometric form of the complex number
z  a  bi is
z  rcos  isin 
where a  r cos , b  r sin , r  a2  b2 ,
and tan  b / a. The number r is the absolute
value or modulus of z, and  is an argument of z.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 6

7. Example Finding Trigonometric
Form
Find the trigonometric form with 0    2 for the
complex number 1 3i.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 7

8. Example Finding Trigonometric
Form
Find the trigonometric form with 0    2 for the
complex number 1 3i.
Find r: r |1 3i | 12  32
 2.
Find  : tan 
3
1
so  

3
.

Therefore, 1 3i  2 cos

3
 isin


3
 
 
.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 8

9. Product and Quotient of Complex
Numbers
Let z1  r1 cos1  isin1   and z2  r2 cos 2  isin 2  .
Then
1. z1  z2  r1r2 cos 1  2   isin 1  2   

.
2.
z1
z2

r1
r2
cos 1  2   isin 1  2    
, r2  0.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 9

10. Example Multiplying Complex
Numbers
Express the product of z1 and z2 in standard form.

z1  4 cos

4
 isin


4
 
 

, z2  2 cos

6
 isin


6
 
 
Copyright © 2011 Pearson, Inc. Slide 6.1 - 10

11. Example Multiplying Complex
Numbers
Express the product of z1 and z2 in standard form.

z1  4 cos

4
 isin


4
 
 

, z2  2 cos

6
 isin
 
z1  z2  r1r2 cos 1  2   isin 1  2   


 4 2 cos

4



6

 
 
 isin

4



6

 
 
 


6

 

 4 2 cos

5
12

 
 
 isin

5
12

 
 
 

 
 
 4 20.259  i0.966 1.464  5.464i
Copyright © 2011 Pearson, Inc. Slide 6.1 - 11

12. A Geometric Interpretation of z2
Copyright © 2011 Pearson, Inc. Slide 6.1 - 12

13. De Moivre’s Theorem
Let z  rcos  isin  and let n be a positive integer.
Then
zn  r cos  isin   



n
 r n cosn  isin n .
Copyright © 2011 Pearson, Inc. Slide 6.1 - 13

14. Example Using De Moivre’s Theorem

Find 
3
2
 i
1
2
 

 
4
using De Moivre's theorem.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 14

15. Example Using De Moivre’s Theorem

Find 
3
2
 i
1
2
 

 
4
using De Moivre's theorem.
The argument of z  
3
2
 i
1
2
is  
7
6
,
and its modulus 
3
2
 i
1
2

3
4

1
4
 1.
Hence,
z  2cos
7
6
 isin
7
6
Copyright © 2011 Pearson, Inc. Slide 6.1 - 15

16. Example Using De Moivre’s Theorem
4
using De Moivre's theorem.

 i
1
2

 
z4  cos 4 

7
6
 
 

 isin 4 

7
6
 
 
 cos

14
3

 
 
 isin

14
3

 
 
 cos

2
3

 
 
 isin

2
3

 
 
 
1
2
 i
3
2

Find 
3
2
 
Copyright © 2011 Pearson, Inc. Slide 6.1 - 16

17. nth Root of a Complex Number
A complex number v  a  bi is an nth root of z if
vn  z.
If z  1, the v is an nth root of unity.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 17

18. Finding nth Roots of a Complex
Number
If z  rcos  isin , then the n distinct
complex numbers

r n cos
  2 k
n
 isin
  2 k
n
 

 ,
where k  0,1,2,..,n 1,
are the nth roots of the complex number z.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 18

19. Example Finding Cube Roots
Find the cube roots of 1.
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20. Example Finding Cube Roots
Find the cube roots of 1.
Write 1 in complex form: z  1 0i  cos0  isin0
The third roots of 1 are the complex numbers
cos
0  2 k
3
 isin
0  2 k
3
for k  0,1,2.
z1  cos0  isin0  1
z2  cos
2
3
 isin
2
3
 
1
2

3
2
i
z3  cos
4
3
 isin
4
3
 
1
2

3
2
i
Copyright © 2011 Pearson, Inc. Slide 6.1 - 20

21. Quick Review
1. Write the roots of the equation x2 12  6x in a  bi form.
2. Write the complex number 1 i3
in standard form a  bi.
3. Find all real solutions to x3  27  0.
Find an angle  in 0    2 which satisfies both equations.
4. sin 
1
2
and cos  
3
2
5. sin  
2
2
and cos  
2
2
Copyright © 2011 Pearson, Inc. Slide 6.1 - 21

22. Quick Review Solutions
1. Write the roots of the equation x2 12  6x in a  bi form.
3 3i, 3 3i
2. Write the complex number 1 i3
in standard form a  bi.
2  2i
3. Find all real solutions to x3  27  0. x  3
Find an angle  in 0    2 which satisfies both equations.
4. sin 
1
2
and cos  
3
2
  5 / 6
5. sin  
2
2
and cos  
2
2
  5 / 4
Copyright © 2011 Pearson, Inc. Slide 6.1 - 22

23. Chapter Test
1. Let u  2, 1 and v  4,2 . Find u v.
2. Let A  (2, 1),B  (3,1),C  (4,2), and D  (1, 5).
Find the component form and magnitude of the vector
uuur
uuur
AC
+BD
3. Given A  (4,0) and B  (2,1), find (a) a unit vector in
uuur
the direction of AB
and (b) a vector of magnitude 3 in
the opposite direction.
4. Given u  4,3 and v  2,5 , find (a) the direction
angles of u and v and (b) the angle between u and v.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 23

24. Chapter Test
5. Convert the polar coordinate (  2.5,25o) to a rectangular
coordinate.
6. Eliminate the parameter t. x  4  t, y  8  5t,  3  t  5.
7. Find a parameterization for the line through the points
( 1, 2) and (3,4).


8. Use De Moivre's theorem to evaluate 3 cos

4
 isin


4
 
 
 

 
5
.
Write your answer in (a) trigonometric form and (b) standard
form.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 24

25. Chapter Test
9. Convert the polar equation r  3cos  2sin to
rectangular form.
10. A 3000 pound car is parked on a street that makes
an angle of 16o with the horizontal.
(a) Find the force required to keep the car from rolling
down the hill.
(b) Find the component of the force perpendicular to
the street.
Copyright © 2011 Pearson, Inc. Slide 6.1 - 25

26. Chapter Test Solutions
1. Let u  2, 1 and v  4,2 . Find u v. 6
2. Let A  (2, 1),B  (3,1),C  (4,2), and D  (1, 5).
Find the component form and magnitude of the vector
uuur
uuur
AC
+BD
8, 3 ; 73
3. Given A  (4,0) and B  (2,1), find (a) a unit vector in
uuur
the direction of AB
and (b) a vector of magnitude 3 in
the opposite direction. (a) 
2
5
,
1
5
(b)
6
5
, 
3
5
Copyright © 2011 Pearson, Inc. Slide 6.1 - 26

27. Chapter Test Solutions
4. Given u  4,3 and v  2,5 , find (a) the direction
angles of u and v and (b) the angle between u and v.

(a) tan1 3
4
 


  0.64 tan1 5
2
 

  1.19 (b)  0.55
5. Convert the polar coordinate (  2.5,25o) to a
rectangular coordinate.  (  2.27, 1.06)
6. Eliminate the parameter t. x  4  t, y  8  5t,
 3  t  5. y  5x 12
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28. Chapter Test
7. Find a parameterization for the line through the points
( 1, 2) and (3,4). x  2t  3, y  3t  4


8. Use De Moivre's theorem to evaluate 3 cos

4
 isin


4
 
 





5
.
Write your answer in (a) trigonometric form and (b) standard
form.

(a) 243 cos
5
4
 isin

5
4
 
  (b)
243 2
2

243 2
2
i
Copyright © 2011 Pearson, Inc. Slide 6.1 - 28

29. Chapter Test
9. Convert the polar equation r  3cos  2sin to

rectangular form. x 
3
2
 

 
2
 y 12

13
4
10. A 3000 pound car is parked on a street that makes
an angle of 16o with the horizontal.
(a) Find the force required to keep the car from rolling
down the hill.  826.91 pounds
(b) Find the component of the force perpendicular to
the street. 2883.79 pounds
Copyright © 2011 Pearson, Inc. Slide 6.1 - 29