Download Plotting Complex Numbers

Chapter 6
Additional Topics
in Trigonometry
6.5 Complex Numbers in
Polar Form; DeMoivre’s
Theorem
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Objectives:
•
•
•
•
Plot complex number in the complex plane.
Find the absolute value of a complex number.
Write complex numbers in polar form.
Convert a complex number from polar to rectangular
form.
• Find products of complex numbers in polar form.
• Find powers of complex numbers in polar form.
• Find roots of complex numbers in polar form.
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The Complex Plane
A complex number z = a + bi is represented as a point
(a, b) in a coordinate plane. The horizontal axis of the
coordinate plane is called the real axis. The vertical axis
is called the imaginary axis. The coordinate system is
called the complex plane.
When we represent a complex
number as a point in the complex
plane, we say that we are
plotting the complex number.
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Example: Plotting Complex Numbers
Plot the complex number in the complex plane: z = 2 + 3i
z  a  bi
z  2  3i
a  2, b  3
z  2  3i
We plot the point
(a, b) = (2, 3).
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Example: Plotting Complex Numbers
Plot the complex number in the complex plane: z = –3 – 5i
z  a  bi
z  3  5i
a  3, b  5
We plot the point
(a, b) = (–3, –5).
z  3  5i
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Example: Plotting Complex Numbers
Plot the complex number in the complex plane: z = –4
z  a  bi
z  4  0i
a  4, b  0
z  4  0i
We plot the point
(a, b) = (–4, 0).
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Example: Plotting Complex Numbers
Plot the complex number in the complex plane: z = –i
z  a  bi
z  0i
a  0, b  1
We plot the point
(a, b) = (0, –1).
z  0i
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The Absolute Value of a Complex Number
The absolute value of the complex number z = a + bi is
the distance from the origin to the point z in the
complex plane.
The absolute value of the complex number a + bi is
z  a  bi  a 2  b 2 .
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Example: Finding the Absolute Value of a Complex
Number
Determine the absolute value of the following complex
z  5  12i
number:
z  5  12i
z  a  bi  a 2  b 2 .
z  5  12i  52  122  25  144  169  13
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Example: Finding the Absolute Value of a Complex
Number
Determine the absolute value of the following complex
z  2  3i
number:
z  2  3i
z  a  bi  a 2  b 2 .
z  2  3i  22  (3) 2  4  9  13
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Polar Form of a Complex Number
A complex number in the form z = a + bi is said to be in
rectangular form.
The expression z  r (cos  i sin  ) is called the
polar form of a complex number.
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Polar Form of a Complex Number (continued)
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Example: Writing a Complex Number in Polar Form
Plot the complex number in the complex plane, then write
the number in polar form: z  1  i 3
z  a  bi
z  1  i 3
a  1, b   3
z  1  i 3
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Example: Writing a Complex Number in Polar Form
Plot the complex number in the complex plane, then write
the number in polar form: z  1  i 3
r  a b
2
2

 (1)   3
2

2
 1 3  4  2
b  3
 3
tan   
1
a
4

3
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Example: Writing a Complex Number in Polar Form
Plot the complex number in the complex plane, then write
the number in polar form: z  1  i 3
4
r  2, 
3
z  r (cos  i sin  )
The polar form of z  1  i 3
4
4 

is z  2  cos
 i sin  .
3
3 

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The Product of Two Complex Numbers in Polar Form
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Example: Finding Products of Complex Numbers in Polar
Form
Find the product of the complex numbers. Leave the
answer in polar form.
z2  5(cos 20  i sin 20)
z1  6(cos 40  i sin 40)
z1 z2  r1r2 [cos(1   2 )  i sin(1   2 )]
z1 z2   6 5[cos(40  20)  i sin(40  20)]
 30(cos60  i sin 60)
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The Quotient of Two Complex Numbers in Polar Form
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Example: Finding Quotients of Complex Numbers in
Polar Form
Find the quotient of the following complex numbers.
Leave the answer in polar form.
4
4 




z1  50  cos
 i sin 
z2  5  cos  i sin 
3
3 
3
3


z1 r1
 [(cos(1   2 )  i sin(1   2 )]
z2 r2
z1 50   4  
4   

 cos 
   i sin 
    10(cos   i sin  )
z2 5   3 3 
 3 3 
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Powers of Complex Numbers in Polar Form
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Example: Finding the Power of a Complex Number
Find [2(cos30  i sin 30)]5 . Write the answer in
rectangular form, a + bi .
z n  r n (cos n  i sin n )
[2(cos30  i sin 30)]5  25 cos  5 30   i sin  5 30 
 32(cos150  i sin150)
 3 1 
 32  
 i
 2 2 
 16 3  16i
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DeMoivre’s Theorem for Finding Complex Roots
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Example: Finding the Roots of a Complex Number
Find all the complex fourth roots of 16(cos60° + isin60°).
Write roots in polar form, with  in degrees.
There are exactly four fourth roots of the given complex
number. The four fourth roots are found by substituting
0, 1, 2, and 3 for k in the expression
  360k  
    360k 

n
zk  r cos 
  i sin 

n
n



 
  60  360 0 
 60  360 0  
4
z0  16 cos 
 i sin 


4
4



 
 2(cos15  i sin15)
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Example: Finding the Roots of a Complex Number
Find all the complex fourth roots of 16(cos60° + isin60°).
  360k  
    360k 

zk  r cos 
  i sin 

n
n



 
n
  60  360 1 
 60  360 1  
z1  16 cos 
 i sin 


4
4



 
4
420
420   2(cos105  i sin105)

 2  cos
 i sin

4
4 

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Example: Finding the Roots of a Complex Number
Find all the complex fourth roots of 16(cos60° + isin60°).
  360k  
    360k 

zk  r cos 
  i sin 

n
n



 
n
  60  360 2 
 60  360 2  
z2  16 cos 
 i sin 


4
4



 
4
780
780   2(cos195  i sin195)

 2  cos
 i sin

4
4 

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Example: Finding the Roots of a Complex Number
Find all the complex fourth roots of 16(cos60° + isin60°).
  360k  
    360k 

zk  r cos 
  i sin 

n
n



 
n
  60  360 3 
 60  360 3  
z3  16 cos 
 i sin 


4
4



 
4
1140
1140  2(cos 285  i sin 285)

 2  cos
 i sin

4
4 

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Example: Finding the Roots of a Complex Number
Find all the complex fourth roots of 16(cos60° + isin60°).
Write roots in polar form, with  in degrees.
The four complex fourth roots are:
z0  2(cos15  i sin15)
z1  2(cos105  i sin105)
z2  2(cos195  i sin195)
z3  2(cos 285  i sin 285)
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