Download Trigonometric Form of Complex Numbers

Trigonometric Form of
Complex Numbers
Lesson 5.2
Graphical Representation of a Complex
Number

Graph in coordinate plane



Called the complex plane
Horizontal axis
is the real axis
Vertical axis is
the imaginary
axis
3 + 4i
•
-2 + 3i
•
•
-5i
2
Absolute Value of a Complex Number

Defined as the length of the line
segment



From the origin
To the point
Calculated by
using Pythagorean
Theorem
3 + 4i
•
3  4i  32  42  25  5
3
Find That Value, Absolutely

Try these


Graph the complex number
Find the absolute value
z  4  4i
z  5
z  5  6i
4
Trig Form of Complex Number


Consider the graphical representation
We note that a right
triangle is formed
a + bi
•
r
b
θ
a
a
cos  
r
a  r cos 
b
sin  
r
b  r sin 
where r  z  a 2  b 2
How do we
determine θ?
b
  tan
a
1
5
Trig Form of Complex Number

Now we use a  r cos 
b  r sin 
and substitute into z = a + bi
z  r  cos  i  r  sin 

Result is

Abbreviation is often
z  r  cis
6
Try It Out

Given the complex number -5 + 6i




Write in trigonometric form
r=?
θ=?
Given z = 3 cis 315°




Write in standanrd form
r=?
a=?
b=?
7
Product of Complex Numbers in Trig
Form

Given
z1  r1  cos1  i  sin 1 

z2  r2  cos2  i  sin 2 
It can be shown that the product is
z1  z2  r1  r2  cis 1  2 


Multiply the absolute values
Add the θ's
8
Quotient of Complex Numbers in Trig
Form

Given
z1  r1  cos1  i  sin 1 

z2  r2  cos2  i  sin 2 
It can be shown that the quotient is
z1 r1
  cis 1   2 
z2 r2
9
Try It Out

Try the following operations using trig
form
 4  cis120    6  cis315 

15cis240
3cis135
Convert answers to standard form
10
Assignment



Lesson 5.2
Page 294
Exercises 1 – 57 odd
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