The Modulus/Argument form of a complex number

The Modulus/Argument form of
a complex number
Given z = x + iy with
z  r and arg(z) = 
y
z  x  iy
P(x, y)
z  r cos   ir sin 
r
z  r cos   i sin  
rsin

0
rcos
x
Example
Express the following in modulus and argument form.
(i)
z  3 i
(ii)
z  1  i
(iii)
z  sin   i cos 
0  

2
(i)
z  3 i
y
z  3 1  2
1
z

3
x
tan  

arg z  
1
3

6

6



z  2 cos  i sin 
6
6

(ii) z  1  i
y
-1
z  11  2

z
x
-1
1
tan  
1


4
3
arg z   
4
  3 
 3
z  2  cos 
  i sin 
 4
  4 

 

(iii)
z  sin   i cos 
y
z  sin 2   cos 2   1
cos
z


sin x
cos 
tan  
sin 
tan  
!
sin 
 tan 
cos 
 
arg z  

2

 



z  1 cos     i sin    

2

 2