Complex
Analysis. FreeTutorial
4.1
Conversion from trigonometric to algebraic
form
4.2
Conversion from algebraic to trigonometric
form
4.3
Examples of the conversion from algebraic
to
trigonometric
form
4.1
The conversion from trigonometric to algebraic
form of the complex number
The conversion from
trigonometric to algebraic form of the
complex number is given by
x = zcos
arg(z),
y = zsin
arg(z).
(1.6)
Example
3:
Trigonometric form:
z
= 4(cos+
i sin).
Algebraic form:
x
= 4cos
= 2,
y
= 4sin
= 2,
z
= 2 + 2i.
Top
4.2
The conversion from algebraic to trigonometric
form of the complex number
The conversion from
algebraic to trigonometric form of the
complex number is given by
z
= ,

(1.7)

It is necessary to be careful
in specifying the choices of
so that the point z
lies in the appropriate quadrant.
Example
4:
Let us consider 4 cases, when the point
z
lies in 4 different quadrants. Let Re(z)
and Im(z)
= ±1.
z 
arg(z) 
Quadrant
1

Formula
for arg(z) 
arg(z)
=;
x>0,
y0 
Result
for arg(z) 
arg(z)
= arctan(1)
= 

Quadrant
2


Quadrant
3


Quadrant
4

Formula
for arg(z) 
arg(z)
=;
x>0,
y0 
Result
for arg(z) 
arg(z)
= arctan(1)
=  

The trigonometric form of
z:
z
= z(cos
Arg(z)
+i
sin
Arg(z)).
Re(z)
and Im(z)
= ±1. z
= =
2
for all cases.
z
= 1 +
i 
z
= (cos
+ i
sin) 
z
= 1 +
i 
z
= (cos+
i
sin) 
z
= 1 
i 
z
= (cos()
+ i
sin()) 
z
= 1 
i 
z
= (cos()
+ i
sin()) 
See Figure 1.5 for all 4
cases of this example.
Figure 1.5
Example of calculating arg(z);
the point z
lies in different quadrants.
You
can see more examples of the conversion
from algebraic to trigonometric form.
Examples
conversion
If z
is purely imaginary number, then x
= 0, and becomes
undefined. We emphasize these special
cases:

(1.8)

Example
5:
z
= i
What is the trigonometric form of
z?
z
= z(cos
arg(z)
+i
sin
arg(z)).
z
= =
1
arg(z)
=
because x
= 0, y
>0.
z
= cos
+ i
sin
If y
= 0, then z
becomes a real number.

(1.9)

Example
6:
z
= 1
What is the trigonometric form of
z?
z
= z(cos
arg(z)
+i
sin
arg(z)).
z
= 1
arg(z)
= 0 because x
>0, y
=0.
z = cos(0)
+ i
sin(0)
= 1.
The algebraic form coincides
with trigonometric one for real numbers.
arg(z)
is indeterminate if x
= 0, y
= 0.
In selecting the proper
values for arg(z),
we must be careful in specifying the choices
of
so that the point
z lies in the appropriate
quadrant.
Examples
of the conversion from algebraic to trigonometric
form
by
Tetyana Butler
