Download Complex Numbers

CHAPTER 8:
Applications of Trigonometry
8.1
8.2
8.3
8.4
8.5
8.6
The Law of Sines
The Law of Cosines
Complex Numbers: Trigonometric Form
Polar Coordinates and Graphs
Vectors and Applications
Vector Operations
Copyright © 2009 Pearson Education, Inc.
8.3
Complex Numbers: Trigonometric Form





Graph complex numbers.
Given a complex number in standard form, find
trigonometric, or polar, notation; and given a complex
number in trigonometric form, find standard form.
Use trigonometric notation to multiply and divide
complex numbers.
Use DeMoivre’s theorem to raise a complex number to
powers.
Find the nth roots of a complex number.
Copyright © 2009 Pearson Education, Inc.
Graphical Representation
Just as real numbers can be graphed on a line, complex
numbers can be graphed on a plane. We graph a
complex number a + bi in the same way that we graph
an ordered pair of real numbers (a, b). However, in
place of an x-axis,we have a real axis, and in place of a
y-axis, we have an imaginary axis. Horizontal
distances correspond to the real part of a number.
Vertical distances correspond to the imaginary part.
Copyright © 2009 Pearson Education, Inc.
Slide 8.3 - 4
Example
Graph each of the following complex numbers.
a) 3 + 2i
b) –4 – 5i c) –3i d) –1 + 3i
Solution:
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e) 2
Slide 8.3 - 5
Absolute Value
The absolute value of a complex number a + bi is
a  bi  a 2  b 2 .
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Slide 8.3 - 6
Example
Find the absolute value of each of the following.
4
b)  2  i
a) 3  4i
c) i
5
Solution:
a) 3  4i  3  4
2
b) 2  i

2
 9 16  25  5
2   1
2
2
 4 1  5
2
2
4
4
4
4
4


2
c)
i  0 i  0      
 5
 5
5
5
5
Copyright © 2009 Pearson Education, Inc.
Slide 8.3 - 7
Trigonometric Notation
If we let  be an angle in
standard position whose
terminal side passes through
the point (a, b), then
a
cos  , or a  r cos
r
b
sin   , or b  r sin 
r
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Slide 8.3 - 8
Trigonometric Notation
Trigonometric Notation for Complex Numbers
a  bi  r cos  i sin 
r is called the absolute value of a + bi.
 is called the argument of a + bi.
This notation is also called polar notation.
To find trigonometric notation for a complex number
given in standard notation a + bi, we must find r
and determine the angle  for which sin  = b/r and
cos  = a/r.
Copyright © 2009 Pearson Education, Inc.
Slide 8.3 - 9
Example
Find trigonometric notation for each of the following
complex numbers.
a) 1 i
b) 3  i
Solution:
a) Note that a = 1 and b = 1. Then
r  a 2  b 2 12  12  2
1
b
2
sin   

r
2
2
1
a
2
cos  

r
2
2
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 in Q I,  = π/4 or 45º



1  i  2  cos  i sin 

4
4
or
1  i  2 cos 45º i sin 45º 
Slide 8.3 - 10
Example
Solution continued
b) a =
3 and b = –1. Then
r  a b 
2
2
 3   1
b
1
1
sin   

r
2
2
a
3
3
cos  

2
r
2
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2
2
2
 in Q IV,  = 11π/6 or 330º
11
11 

3  i  2  cos
 i sin


6
6
or
3  i  2 cos 330º i sin 330º 
Slide 8.3 - 11
Example
Find standard notation, a + bi, for each of the
following complex numbers.
7
7 

b) 8  cos
 i sin 
a) 2 cos120º i sin120º 

4
4 
Solution:
a) 2 cos120º i sin120º   2 cos120º  2sin120º i
 1
a  2cos120º  2    
 2
 1
 3
b  2sin120º  2  
 3

 2 
2 cos120º i sin120º   1  3i
Copyright © 2009 Pearson Education, Inc.
Slide 8.3 - 12
Example
Solution continued
b)
7 
7 
7
7 

  8 sin  i
8  cos
 i sin   8 cos

4 
4 
4
4 
 2
7
a  8 cos
 8 
2

4
 2 

2
7
 2
b  8 sin
i  8  

4
 2 
7
7 

8  cos
 i sin   2  2i

4
4 
Copyright © 2009 Pearson Education, Inc.
Slide 8.3 - 13
Complex Numbers: Multiplication
Complex Numbers: Multiplication
For any complex numbers r1(cos 1 + i sin 1) and,
r1(cos 1 + i sin 1),
r1 cos1  i sin 1  r2 cos 2  i sin  2 
 r1r2  cos 1   2  i sin 1   2  .
Copyright © 2009 Pearson Education, Inc.
Slide 8.3 - 14
Example
Convert to trigonometric notation and multiply.
1  i 
3i

Solution:
First find trigonometric notation, then multiply.
1  i  2 cos 45º i sin 45º 
3  i  2 cos 330º i sin 330º 
2 cos 45º i sin 45º  2 cos 330º i sin 330º 
 2 2  cos 45º 330º   i sin 45º 330º 
 2 2 cos 375º i sin 375º 
 2 2 cos15º i sin15º 
Copyright © 2009 Pearson Education, Inc.
Slide 8.3 - 15
Complex Numbers: Division
Complex Numbers: Division
For any complex numbers r1(cos 1 + i sin 1) and,
r1(cos 1 + i sin 1), r2 ≠ 0,
r1 cos1  i sin1  r1
  cos 1   2  i sin 1   2  .
r2 cos 2  i sin 2  r2
Copyright © 2009 Pearson Education, Inc.
Slide 8.3 - 16
Example
3
3 




 i sin  by 4  cos  i sin 
Divide 2  cos


2
2 
2
2
and express the answer in standard notation.
Solution:
3
3 

2  cos
 i sin 

2   3  
 3   
2
2 
  cos 
   i sin 
 
 2 2


4   2 2

4  cos  i sin 

2
2
1
 cos   i sin  
2
1
1
 1  i  0   
2
2
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Slide 8.3 - 17
Powers of Complex Numbers
DeMoivre’s Theorem
For any complex number r(cos  + i sin ) and any
natural number n,
 r cos  i sin    r n cos n  i sin n .
n
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Slide 8.3 - 18
Example
Find each of the following. a) 1 i 
9
b)
 3  i
10
Solution:
First find trigonometric notation, then raise to the power.
1  i  
1 i 
9
2 cos 45º i sin 45º 
  2 cos 45º i sin 45º 

9
 2  cos9  45º  i sin 9  45º 
2
9
92
 16
cos 405º i sin 405º 
2 cos 45º i sin 45º   16
 16 16i
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 2
2
2
i
2 
 2
Slide 8.3 - 19
Example
Solution continued
b)
 3  i
10
First find trigonometric notation, then raise to the power.
 3  i  2 cos 330º i sin 330º 
 3  i   2 cos 330º i sin 330º 
10
10
 210 cos 3300º i sin 3300º 
 1024 cos60º i sin 60º 
1
3
 1024   i
 512  512 3i

2 
2
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Slide 8.3 - 20
Roots of Complex Numbers
Roots of Complex Numbers
The nth roots of a complex number r(cos  + i sin )
r ≠ 0, are given by
r
1n
360º 
360º  
 

 cos  n  k  n   i sin  n  k  n   ,


where k = 0, 1, 2, …, n –1.
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Slide 8.3 - 21
Example
Find the cube roots of 1. Then locate them on a graph.
Solution:
First find trigonometric notation, then raise to the power.
1  1cos0º i sin 0º 
Then n = 3, 1/n = 1/3, and k = 0, 1, 2; so,
1cos 0º i sin 0º 
13
360º 
360º  
  0º
 0º
 1  cos   k 
 i sin   k 
, k  0,1,2



 3
3 
3 
  3
13
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Slide 8.3 - 22
Example
Solution continued
The roots are
1cos0º i sin 0º   1
1
3
i
1cos120º i sin120º    
2 2
1
3
i
1cos240º i sin 240º    
2 2
The graphs of the cube roots lie equally spaced about a
circle of radius 1. The roots are 360º/3, or 120º apart,
as shown on the next slide.
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Slide 8.3 - 23
Example
Solution continued
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Slide 8.3 - 24