Download Trigonometric Form of Complex Numbers

Chapter 6
Additional Topics in Trigonometry
6.5 Trig Form of a Complex Number
Objectives:
 Find absolute values of complex numbers.
 Write trig forms of complex numbers.
 Multiply and divide complex numbers written in
trig form.
 Use DeMoivre’s Theorem to find powers of
complex numbers.
 Find nth roots of complex numbers.
2
Graphical Representation of a
Complex Number
 Graph in coordinate plane called the complex plane


Horizontal axis
is the real axis.
Vertical axis is
the imaginary
axis.
3 + 4i
•
-2 + 3i
•
•
-5i
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Absolute Value of a
Complex Number
 Defined as the length of the line segment from the
origin (0, 0) to the point.
 Calculate using the Distance Formula.
3 + 4i
•
z  a  bi  a 2  b2
3  4i  3  4  25  5
2
2
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Examples
 Graph the complex number.
 Find the absolute value.
z  5
z  4  4i
z  5  6i
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Trig Form of Complex Number
 Graph the complex number.
 Notice that a right triangle is formed.
a
cos  
r
a  r cos 
b
sin  
r
b  r sin 
a + bi
•
r
b
θ
a
where r  z  a 2  b 2
How do we
determine θ?
b
  tan
a
1
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Trig Form of Complex Number
 Substitute a  r cos  &
into z = a + bi.
b  r sin 
z  r  cos  i  r  sin 
z  r cos  i sin  
 Sometimes abbreviated as z  r  cis
 Result is
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Examples
 Write the complex number –5 + 6i in trig form.
r=?
θ=?
 Write z = 3 cos 315° + 3i sin 315° in standard form.
r=?
a=?
b=?
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Product of Trig Form of Complex Numbers
 Given
and
 It can be shown that the product is
 That is,
 Multiply the absolute values.
 Add the angles.
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Quotient of Trig Form of Complex Numbers
 Given
and
 It can be shown that the quotient is
 That is,
 Divide the absolute values.
 Subtract the angles.
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Examples
 Calculate using trig form and convert answers to
standard form.
1.
4 cos120


 4i sin 120 6 cos 315  6i sin 315

15 cos 240  15i sin 240
2.
3 cos 35  3i sin 35
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Powers of Complex Numbers
 If z = r (cos θ + i sin θ), find z2.
z 2  r  cos   i  sin    r  cos   i  sin  
 r 2   cos 2  i  sin 2 
 What about z3?
12
DeMoivre’s Theorem
 If z = r (cos θ + i sin θ) is a complex number
and n is a positive integer, then
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Examples
 Apply DeMoivre’s Theorem.
1.
3 cos 330

 i sin 330


4
12

2
2

2.  
i

2
2


14
Roots of Complex Numbers
 Recall the Fundamental Theorem of Algebra in
which a polynomial equation of degree n has
exactly n complex solutions.
 An equation such as x6 = 1 will have six solutions.
Each solution is a sixth root of 1.
 In general, the complex number u = a + bi is an
nth root of the complex number z if
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Solutions to Previous Example
 An equation such as x6 = 1 will have six solutions.
Each solution is a sixth root of 1.
16
th
n
Roots of a Complex Number
 For a positive integer n, the complex number
z = r (cos θ + i sin θ) has exactly n distinct nth
roots given by
 Note: The roots are equally spaced around a
circle of radius n r centered at the origin.
17
Example
 Find the three cube roots of z = –2 + 2i.
 Write complex number in trig form.
 Find r.
 Find θ.
 Use the formula with k = 0, 1, and 2.
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Solution
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Homework 6.5
 Worksheet 6.5
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