Download Section 6.5 - Shelton State

Section 6.5
Complex Numbers in Polar Form
Overview
• Recall that a complex number is written in the
form a + bi, where a and b are real numbers
and
i  1
• While it is not possible to graph complex
numbers on a real number plane, a similar
setup can be used.
The Complex Plane
Graph Each of the Following
• z = 3i
• z = -5 + 2i
• z = 3 – 4i
Absolute value of a complex
number
• The absolute value of a complex number z is
the distance from the origin to the point z in
the complex plane:
z  a b
2
2
Polar form of a complex number
• When a complex number is in a + bi form,
it is said to be in rectangular form.
• Just as we superimposed the polar plane
onto the rectangular coordinate plane, we
can do the same thing with the complex
plane.
Continued
z  r (cos   i sin  )
a  r cos 
b  r sin 
r  a2  b2
b
tan   ,0    2
a
•r is called the modulus and “theta” is called
the argument.
Examples
• Graph each of the following, then write the
complex number in polar form:
1  i
5
3 3  3i
Now, the Other Way
• Write each complex number in rectangular
form:
3cos 330  i sin 330



4 cos  i sin 
4
4

Products and Quotients
• Given z1  r1 cos 1  i sin 1 
z2  r2 cos  2  i sin  2 
, two complex
numbers in polar form.
• Their product and quotient can be found by the
following:
z1 z 2  r1r2 cos1   2   i sin 1   2 
z1 r1
 cos1   2   i sin 1   2 
z 2 r2
In Other Words…
• When multiplying, multiply the moduli and
add the arguments.
• When dividing, divide the moduli and
subtract the arguments.
• Keep in mind that you may have to rename your argument so that is an angle
between 0 and 360° or 0 and 2π radians.
Raising to a Power
When raising a complex number
to a power, use DeMoivre’s
Theorem:
n
n
z  r cos n  i sin n 
In other words, raise the
modulus to the nth power and
multiply the argument by n
(again, be prepared to
rename your argument).
A Final Word Before the
Examples
• Pay particular attention to the form your
final answer should take (complex polar or
complex rectangular).
Find the Product (Answer in
Polar Form)
z1  cos
z 2  cos

4

6
z1  2  2i
z 2  1  i
 i sin
 i sin

4

6
Find the Quotient z1/z2(Answer
in Polar Form)
z1  72cos 12  i sin 12
z 2  9cos 4  i sin 4

 

z1  4 cos  i sin 
10
10 


 

z 2  9 cos  i sin 
12
12 

Use the French Guy’s Theorem (write
answers in rectangular form)
6cos15  i sin 15
3
 
2
2 
3
cos

i
sin



3
3 
 
8
Finding Complex Roots
Let w = r(cos θ + i sin θ) be a complex
number in polar form. w has n distinct
complex nth roots given by
    2k 
   2k 
zk  r cos
  i sin 
, k  0,1,2,..., n  1
 n 
  n 
    360k 
   360k 
zk  n r cos

i
sin


, k  0,1,2,..., n  1
n
n



 
n
Examples
• Find all the complex cube roots of 8. Write
your answers in rectangular form.
• Find all the complex fourth roots of
16(cos120° + I sin120°). Write your
answers in polar form.