Download Complex Numbers in Trigonometric Forms

Complex Numbers in Polar Form
Objectives of this Section
• Convert a Complex Number from Rectangular Form
to Polar Form
• Plot Points in the Complex Plane
• Find the Products and Quotients of Complex
Numbers in Polar Form
• Use De Moivre’s Theorem
• Find Complex Roots
The Complex Plane
Imaginary
Axis
O
z  x  yi
Real
Axis
Let z  x  yi be a complex number. The
magnitude or modulus of z, denoted by
z is defined as the distance from the origin
to the point ( x, y). Thus,
z x y
2
2
Imaginary
Axis
y
z  x  yi
z
O
x
Real
Axis
Recall, if z  x  yi , then its conjugate,
denoted by z , is z  x  yi. Because
zz  x  y , it follows that:
2
2
z  zz
z  x  yi  r cos r sini

Cartesian
Form

Polar Form
z  rcos  i sin
where  , 0   < 2 , is the argument of z
z  zz 

rcos  i sinrcos i sin

 r cos  icossin isincos i sin 
2
2
2
 r cos   sin   r
2
2
z r
2
Plot the point corresponding to z  3  4i in
the complex plane, and write an expression
for z in polar form.
Imaginary
4
Axis
z  3  4i
-3
Real Axis
z   3  4i
Quadrant II
x  3 and y  4
r  ( 3)  4  9  16  5
y 4
sin   
0   < 2
r 5
2
2
  180  53.1  126.9







z r cos
 isin 5cos126
.9 isin126
.9 
Write an expression for

z  3 cos 330  i sin 330
in rectangular form.

z  3 cos 330  i sin 330
 3 1 
 3
 i 
 2 2 


Theorem
Let z1  r1 cos1  i sin 1  and
z2  r2 cos2  i sin 2  be two complex
numbers. Then
z1z2  r1r2 cos1  2   i sin1  2 
If z 2  0 , then
z1 r1
 cos1  2   i sin1  2 
z2 r2


 i sin 120 , find
If z  4 cos 40  i sin 40 and

w  6 cos120
(a) zw

(b) z w
 

zw 4 cos40 i sin40 6 cos120 i sin120



 




 4  6 cos 40 120  i sin 40 120



 24 cos160  i sin160








w  6cos120  i sin 120 
4cos 40  i sin 40 
z

w 6cos120  i sin 120 
4
 cos40  120  i sin40
6
z  4 cos 40  i sin 40
 



  


2
 cos  80  i sin  80
3
2
 cos 280  i sin 280
3
 120

DeMoivre’s Theorem
If z  rcos  i sin is a complex number,
then
z  r cos n  i sin n 
n
n
where n  1 is a positive integer.

Write 3 cos 30  i sin 30

4
in the
standard form a  bi .
3cos 30
 i sin 30

4
 


 81cos120  i sin120 
 3 cos 4  30  i sin 4  30
4

 1
3 
81 81 3
 81   
i   
i
2
2
 2 2 


Write  3  i in the standard form a  bi.
4
r


 3
2
1  4  2
2
5
5 


3 1 
 i sin

3  i  2  
 i   2 cos
6
6 
2
2 



3i

4
 
5
5  
  2  cos
 i sin

6
6 
 
4

 5 
 5
 2  cos  4 
  i sin  4 
6 
6



4




3  i
4
  5
 2  cos  4 
6
 

 5
  i sin  4 
6





 10  
 10 

  i sin 
 3 
 3



4

 16  cos

 1 
 
3
i 
 16     
 
 2
2

 

  8  8 3i
Let w be a complexnumberand let n  2 denote
a positiveinteger. Any complexnumber that
satisfiesthe equation
zn  w
is called the complexnth rootof w.
Finding Complex Roots
Let w  rcos  i sin   be a complexnumber.
If w  0, thereare n distinct complexnth roots
of w, given by theformula
   2 k 
  2 k  
z k  r  cos  
  i sin  

n 
n 
n
 n
n
where k  0, 1, 2,
, n 1
Find the complex fifth roots of  2 3  2i.

3 




 2 3  2i  2 
 i   2 cos 150  i sin 150
 2


The five complex roots of  2 3  2i are








150 360 k
150 360 k 
5
  i sin

zk  2 cos


5 
5 
  5
 5
 


zk  5 2 cos 30  72 k  i sin 30  72 k
k  0, 1, 2 , 3, 4


 


zk  2 cos 30  72 k  i sin 30  72 k
k  0, 1, 2 , 3, 4
z0  5 2 cos 30  i sin 30
5
z1  5
z2  5
z3  5
z4  5


2cos102  i sin 102 
2cos174  i sin 174 
2cos 246  i sin 246 
2cos 318  i sin 318 
