CHAPTER 4.3
CHAPTER 4 COMPLEX NUMBERS
PART 3 –Trigonometric Form of a Complex Number
TRIGONOMETRY MATHEMATICS CONTENT STANDARDS:
 17.0 - Students are familiar with complex numbers. They can represent a
complex number in polar form and know how to multiply complex
numbers in their polar form.
OBJECTIVE(S):
 Students will learn the definition of the absolute value of a complex
number and how to find it.
 Students will learn how to write a complex number in trigonometric
form.
 Students will learn how to write a complex number in standard form.
 Students will learn how to multiply and divide complex numbers.
The Complex Plane
You can represent a complex number z = a + bi as the point (a, b) in a coordinate plane
(the complex plane). The horizontal axis is called the real axis and the vertical axis is
called the imaginary axis.
Imaginary
Axis
5i
4i
3i
2i
3+i
1i
-4
-3
-2
-1-1i
1
2
3
4
5 Real
axis
-2i
-2 - i
-3i
-4i
The absolute value of a complex number a + bi is defined as the distance between the
origin (0, 0) and the point (a, b).
CHAPTER 4.3
Definition of the Absolute Value of a Complex Number
The absolute value of the complex number z = a + bi is
a  bi  ____________
If the complex number a + bi is a real number (that is, if b = 0), then this definition
agrees with that given for the absolute value of a real number
a  0i =
=
EXAMPLE 1: Finding the Absolute Value of a Complex Number
Plot z = -2 + 5i and find its absolute value.
Imaginary
Axis
7i
6i
5i
4i
3i
2i
1i
-5
-4
-3
-2
-1-1i
-2i
z
=
=
1
2
3
4
5 Real
Axis
CHAPTER 4.3
1.) Plot the complex number and find its absolute value.
a. –5 + 2i
b. -3i
Imaginary
Axis
5i
4i
3i
2i
1i
-5
-4
-3
-2
-1-1i
-2i
-3i
-4i
-5i
1
2
3
4
5 Real
Axis
CHAPTER 4.3
Trigonometric Form of a Complex Number
In Section 4.1 you learned how to add, subtract, multiply, and divide complex numbers.
To work effectively with powers and roots of complex number, it is helpful to write
complex numbers in trigonometric form. Consider the nonzero complex number a + bi.
By letting  be the angle from the positive x-axis (measured clockwise) to the line
segment connecting the origin and the point (a, b), you can write
Imaginary
Axis
(a, b)
b
r

a
sin  
cos  
a  r cos
and
Real
Axis
b  r sin 
where r  a 2  b2 . Consequently, you have
a  bi  r cos   r sin  i
from which you can obtain the trigonometric form of a complex number.
Trigonometric Form of a Complex Number
The trigonometric form of the complex number z = a + bi is
z  r cos  i sin  
where a  r cos , b  r sin  , r  a 2  b2 , and tan  
of z, and  is called an argument of z.
b
. The number r is the modulus
a
CHAPTER 4.3
The trigonometric form of a complex number is also called the polar form. Because there
are infinitely many choices for  , the trigonometric form of a complex number is not
unique. Normally,  is restricted to the interval 0    2 , although on occasion it is
convenient to use   0 .
CHAPTER 4.3
EXAMPLE 2: Writing a Complex Number in Trigonometric Form.
Write the complex number z  2  2 3i in trigonometric form.
The absolute value of z is
r=
=
=
=
3
Real
Axis
and the  is
tan  
b
a
=
=
Imaginary
Axis
-5
-4
-3
-2
-1
-1i
1
2
-2i
-3i
z  2  2 3i
-4i
z  2  2 3i lies in Quadrant ______, choose  = _____________. So, the
trigonometric form is
z
=
=
r cos  i sin  
CHAPTER 4.3
EXAMPLE 3: Writing a Complex Number in Trigonometric Form
Write the complex numbering trigonometric form.
z  6  2i
The absolute value of z is
r
=
=
=
=
and the angle  is
tan  =
=
=
Imaginary Axis
4i
3i
z  6  2i
2i
1i
-2
-1
-1i
1
2
3
4
5
6
-2i
Because z  6  2i is in Quadrant ____, you can conclude that
7 Real Axis
CHAPTER 4.3

=


So, the trigonometric form of z is
=
=

EXAMPLE 4: Writing a Complex Number in Standard Form
Write the complex number in standard form a + bi.
  
  
z  8  cos    i sin    
 3 
  3
 
 
Because cos   = ________ and sin    = _____, you can write
 3
 3
  
  
8  cos    i sin    
z
=
 3 
  3
=
=
=
DAY 1
Multiplication and Division of Complex Numbers
The trigonometric form adapts nicely to multiplication and division of complex numbers.
Suppose you are given two complex numbers
z1  r1 cos1  i sin 1 
and
z2  r2 cos2  i sin 2 
CHAPTER 4.3
The product of z1 and z2 is
z1 z2  r1r2 cos1  i sin 1 cos2  i sin 2 
= r1r2 cos1 cos2  sin 1 sin 2   isin 1 cos2  cos1 sin 2 
Using the sum and difference formulas for cosine and sine, you can rewrite this equation
as
z1 z2  r1r2 cos1  2   i sin 1  2 
Product and Quotient of Two Complex Numbers
Let z1  r1 cos1  i sin 1  and z2  r2 cos2  i sin 2  be complex numbers
z1 z2  r1r2 cos1  2   i sin 1  2 
z1 r1
 cos1   2   i sin 1   2 , z 2  0
z 2 r2
Product
Quotient
Note that this rule says that to multiply two complex numbers you multiply moduli and
add arguments, whereas to divide two complex numbers you divide moduli and subtract
arguments.
EXAMPLE 5: Dividing Complex Numbers in Trigonometric Form
z
Find the quotient 1 of the complex numbers.
z2

z1  24 cos 300 0  i sin 300 0
z1
z2


z 2  8 cos 750  i sin 750

=
=
=
=
=
Divide moduli and
subtract arguments.
CHAPTER 4.3
EXAMPLE 6: Multiplying Complex Numbers in Trigonometric Form
Find the product of the following complex numbers.
2
2 

z1  2 cos
 i sin

3
3 

z1 z2
11
11 

z 2  8 cos
 i sin

6
6 

=
=
Multiply moduli and
add arguments.
=
=
=
=
You can check this result by first converting to the standard forms z1 =
___________________ and z2 = ________________________ and then multiplying
algebraically, as in Section 4.1.
z1 z2
=
=
=
CHAPTER 4.3
2.) Represent the complex number graphically and find the trigonometric form of the
number.

a. 3 3  i

Imaginary
Axis
6i
5i
4i
3i
2i
1i
-5
-4
-3
-2
-1
-1i
-2i
-3i
-4i
-5i
-6i
1
2
3
4
5 Real
Axis
CHAPTER 4.3
b.  1 3i
Imaginary
Axis
6i
5i
4i
3i
2i
1i
-5
-4
-3
-2
-1
-1i
-2i
-3i
-4i
-5i
-6i
1
2
3
4
5 Real
Axis
CHAPTER 4.3
3.) Represent the complex number graphically and find the standard form of the number.

3 cos 2250  i sin 2250

Imaginary
Axis
6i
5i
4i
3i
2i
1i
-5
-4
-3
-2
-1
-1i
-2i
-3i
-4i
-5i
-6i
DAY 2
1
2
3
4
5 Real
Axis