Transcript Document

Trigonometric
(Polar) Form
of Complex
Numbers
How is it Different?
In a rectangular system, you go
left or right and up or down.
z  2  2i
In a trigonometric or polar system,
you have a direction to travel and a
distance to travel in that direction.
z  2  cos 45  i sin 45
Polar form (2,45)
Remember a complex number has a real part and an
imaginary part. These are used to plot complex
numbers on a complex plane.
z  a  bi
z  a b
2
Imaginary
Axis
2
z  a  bi
z

a
b
Real
Axis
The absolute value or
modulus of z denoted by z
is the distance from the
origin to the point (a, b).
The angle formed from the
real axis and a line from the
origin to (a, b) is called the
argument of z, with
requirement that 0   < 2.
b
  tan  
a
1
modified for quadrant
and so that it is
between 0 and 2
Trigonometric Form of a Complex Number
z  r  cos  i sin  
The modulus is r  a  b
2
2
The argument  can be found
r

a
b
b
by using tan   adjusting for
a
correct quadrant if necessary
1
Note: You may use any other trig
functions and their relationships to the
right triangle as well as tangent.
Plot the complex number and then convert to trigonometric
form:
z   3 i
Imaginary
Axis
1
Find the modulus r
r
́
 3

r
Real
Axis
 3   1
2
2
 4 2
Find the argument 
 1 
  tan 
 but in Quad II
 3
1
5
5 

z  2 cos  i sin

6
6 

5

6
It is easy to convert from trigonometric to rectangular
form because you just work the trig functions and
distribute the r through.
5
5   3 1 

z  2 cos  i sin
 i    3  i
  2 
6
6   2 2 

1
2
3

2
1
2
 3
5
6
If asked to plot the point and it is
in trigonometric form, you would
plot the angle and radius.
Notice that is the same as
plotting
 3 i
Graphing Utility:
Write the complex number 3.75 cos 3  i sin 3
4
4
in standard form a + bi.


[2nd] [decimal point]

3.75 cos 3  i sin 3
4
4
 2.652  2.652i
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

7
Multiplying Complex Numbers
To multiply complex numbers in
rectangular form, you would
FOIL and convert i2 into –1.
To multiply complex numbers in trig
form, you simply multiply the rs and
and the thetas.
 a  bi  c  di 
 r1  cos 1  i sin 1    r2  cos 2  i sin 2 
r1r2  cos 1  2   i sin 1  2  
ac  adi  bci  dbi 2
ac  adi  bci  db
 ac  db    ad  bc  i
The formulas are scarier than they are.
Let's try multiplying two complex numbers in trigonometric
form together.
z1  r1 cos1  i sin 1 
z2  r2 cos 2  i sin  2 
z1 z2   r1  cos 1  i sin 1    r2  cos  2  i sin  2  



Look at where
andwhere
we
ended
up and
 r1r2 we
cosstarted
1  i sin
cos


i
sin

1
2
2
see if you can make a statement
as to what
happens
to
Must
FOIL
these two complex
the r 's and the  's when
you
multiply
 numbers.
r1r2 cos 1 cos  2  i sin  2 cos 1  i sin 1 cos  2  i 2 sin 1 sin  2

Replace i 2 with -1 and group real terms and then imaginary terms
Multiply the Moduli and Add the Arguments
 r1r2 cos1 cos2  sin 1 sin 2   sin 1 cos2  cos1 sin 2 i
use sum formula for cos
use sum formula for sin
 r1r2 cos1  2   i sin1  2 

multiply z1z2
Example
Where z1  2 3  2i  4  cos30  i sin 30 
z2  3 2  3 2i  6  cos 45  i sin 45
Rectangular form
z1 z2
2

3  2i 3 2  3 2i
Trig form
z1 z2

6 6  6 6i  6 2i  6 2i 2
 4  cos 30  i sin 30   6  cos 45  i sin 45 
4  6  cos  30  45  i sin  30  45 
6 6  6 6i  6 2i  6 2
24  cos 75  i sin 75
6
r
 

6 6 2  6 6 6 2 i
6
6 6 2
  6
2
6 6 2

2
r  216  72 12  72  216  72 12  72
r  576  24
6 6 6 2 
  75
6 6 6 2 
  tan 1 
Dividing Complex Numbers
In rectangular form, you
rationalize using the complex
conjugate.
a  bi
c  di
 a  bi   c  di 



 c  di   c  di 
ac  adi  bci  bdi 2
c 2  d 2i 2
ac  adi  bci  bd
c2  d 2
ac  bd  bc  ad 
 2
i
2
2
2 
c d
 c d 
In trig form, you just divide the rs
and subtract the theta.
r1  cos 1  i sin 1 
r2  cos  2  i sin  2 
r1
cos 1   2   i sin 1   2  

r2
Let z1  r1 cos 1  i sin 1  and z 2  r2 cos  2  i sin  2 
be two complex numbers. Then
z1 z2  r1r2 cos1  2   i sin1  2 
(This says to multiply two complex numbers in polar
form, multiply the moduli and add the arguments)
If z 2  0, then
z1 r1
 cos1  2   i sin1  2 
z2 r2
(This says to divide two complex numbers in polar form,
divide the moduli and subtract the arguments)




If z  4 cos 40  i sin 40 and w  6 cos 120  i sin 120 ,
find : (a) zw
(b) z w




 


zw   4 cos 40  i sin 40   6 cos120  i sin120 

 


 4  6 cos 40 120  i sin 40 120
multiply the moduli

add the arguments
(the i sine term will have same argument)
 24  cos160   i sin160  
 24  0 .93969  0 .34202 i 
  22 . 55  8 . 21i
If you want the answer
in rectangular
coordinates simply
compute the trig
functions and multiply
the 24 through.




4 cos 40  i sin 40
z

w 6 cos120  i sin 120
 


4
 cos 40  120  i sin 40  120
6
divide the moduli
 
2
 cos  80
3
subtract the arguments
 i sin 80 
  

2
 cos 280  i sin 280
3
In rectangular
coordinates:


In polar form we
want an angle
between 0 and
360° so add
360° to the -80°
2
 0 .1736  0 .9848i   0 .12  0 .66i
3
divide
Example
z1
z2
Where z1  3 2  3 2i  6  cos 45  i sin 45
z2  2 3  2i  4  cos 30  i sin 30 
Rectangular form
Trig form
3 2  3 2i
2 3  2i
6  cos 45  i sin 45
4  cos 30  i sin 30 
 3 2  3 2i   2 3  2i 



2
3

2
i
2
3

2
i



6 6  6 2i  6 6i  6 2i 2
12  4i 2
6 6  6 2i  6 6i  6 2
12  4
6
6 6 2
16
  6
6
cos  45  30   i sin  45  30  

4
3
 cos15  i sin15
2

6 6 2 i
16




 6 6 6 2 


1 
16
  15
  tan
 6 6 6 2 


216  72 12  72  216  72 12  72
16


r


2


 6 6 6 2   6 6 6 2 
 

r 
16
16

 


 

256
r
576
9 3


256
4 2
2
Powers of Complex Numbers
This is horrible in rectangular
form.
 a  bi 
 a  bi  a  bi  a  bi  ...  a  bi 
n
The best way to expand one
of these is using Pascal’s
triangle and binomial
expansion.
You’d need to use an ichart to simplify.
It’s much nicer in trig form. You
just raise the r to the power and
multiply theta by the exponent.
z  r  cos   i sin  
z n  r n  cos n  i sin n 
Example
z  5  cos 20  i sin 20 
z 3  53  cos3  20  i sin 3  20 
z 3  125  cos60  i sin 60 
Roots of Complex Numbers
• There will be as many answers as the index
of the root you are looking for
– Square root = 2 answers
– Cube root = 3 answers, etc.
• Answers will be spaced symmetrically
around the circle
– You divide a full circle by the number of
answers to find out how far apart they are
General Process
1. Problem must be in trig form
2. Take the nth root of n. All answers have
the same value for n.
3. Divide theta by n to find the first angle.
4. Divide a full circle by n to find out how
much you add to theta to get to each
subsequent answer.
The formula
z  r  cos   i sin  
n
  360k
  360k 

z  n r  cos
 i sin
 or
n
n


n
  2 k
  2 k 

r  cos
 i sin

n
n 

k starts at 0 and goes up to n-1
This is easier than it looks.
Example
Find the 4th root of z  81cos80  i sin80
1. Find the 4th root of 81
r  4 81  3
2. Divide theta by 4 to get
the first angle.

3. Divide a full circle (360) by
4 to find out how far apart
the answers are.
4.
List the 4 answers.
80
 20
4
360
 90 between answers
4
z1  3  cos 20  i sin 20 
The only thing that changesz2  3  cos  20  90   i sin  20  90    3  cos110  i sin110 
is the angle.
z  3  cos 110  90  i sin 110  90    3  cos 200  i sin 200 
•
3
•
The number of answers z  3 cos 200  90  i sin 200  90  3 cos 290  i sin 290


 

 
equals the number of roots. 4