Complex
Analysis. FreeTutorial
Theory: Conversion from algebraic to trigonometric
form
Formulas
of the conversion from algebraic to trigonometric
form
Examples
of the conversion from algebraic to trigonometric
form
Formulas of 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)

If z
is purely imaginary number, then x
= 0, and becomes
undefined. We emphasize these special
cases:
(1.8)
If y
= 0, then z
becomes a real number.
(1.9)
arg(z)
is indeterminate if x
= 0, y
= 0.
Top
Examples
of the conversion from algebraic to trigonometric
form
Example
4:
Algebraic form of z
:
z
= 2 + 2i.
What is the trigonometric form of z
?
z
= z{cos
Arg(z)
+i
sin
Arg(z)}
.
z
= =
4.
arg(z)
=
because x>0.
See (1.7)
The principal value arg(z)
= arctan
=.
Trigonometric form:
z
= 4(cos+
i sin).
Arg(z)
= arg(z)
+n
=
+n.
z
= 4(cos(+n)
+ i
sin(+n)).
Example
5:
Algebraic form of z
:
z
= 1 +
i.
What is the trigonometric form of z
?
z
= z{cos
arg(z)
+i
sin
arg(z)}
.
z
= =
2.
arg(z)
=+
because y>0.
See (1.7)
arg(z)
= +
=
Trigonometric form:
z
= 2(cos+
i
sin).
Example
6:
Algebraic form of z
:
z
=
 i.
What is the trigonometric form of z
?
z
= z{cos
arg(z)
+i
sin
arg(z)}
.
z
= =
2.
arg(z)
=
because x>0.
See (1.7)
arg(z)
=
= .
z
= 2(cos()
+ i
sin()).
Example
7:
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()
Example
8:
Algebraic form of z
:
z
= 5 
3i.
What is the trigonometric form of z
?
z
= z{cos
arg(z)
+i
sin
arg(z)}.
z
= =
.
arg(z)
=

because x<0,
y<0.
See (1.7)
arg(z)
=
Trigonometric form:
z
=(cos()
+ i
sin()).
Example
9:
This example is more complicated.
Write a number z
= sin+
i cos
in a form
z = z{cos
arg(z)
+i
sin
arg(z)}.
z
= =
=
1.
arg(z)
=
+
because x<0,
y>0.
See (1.7)
x
=  sin; 
y
= cos;

arg(z)
= arctan{cos()/(sin()}
+
=
+
= .
z
= cos
+i
sin.
We can get the same result geometrically.
Let us represent 2 points in the complex
plain:
a given point z
= sin+
i cos
,
and a point z1
= cos+
i sin.
It is clear that arg(z)
=
=
,
so z
= cos
+i
sin.
Figure 1.5 Example z
= sin+
i cos
by Tetyana Butler
