Download Complex Numbers

Lesson 6.5
Trigonometric Form of Complex
Numbers
Complex Numbers
Recall complex numbers can be written in the form:
z  a  bi
Also, they are graphed on the “real – imaginary” plane:
z  3 2i
Imaginary Axis
Absolute Value of a Complex
Distance from (0,0) to (a, b)
Real
Axis
a  bi  a 2  b2
Trigonometric Form
Remember changing vector components using
trig:
xv , yv  v cos  v , v sin  v
We can do the same with complex numbers
(where r represents the magnitude)
a  bi  r cos  r sin  i
Where r is the distance (absolute
value), so…
r  a 2  b2
Example
Write the complex number in trigonometric form.
z  1 3i
Example
Write the standard form (a + bi) of the complex number

z  5 cos135  i sin 135
o
o

Problem Set 6.5.1
Product & Quotient of Complex Numbers
Start with 2 complex numbers:
z1  r1 cos1  r1 sin 1 i 
z2  r2 cos2  r2 sin 2 i 
Then the product is:
r = magnitude
z1 z2  r1r2 cos1  2   r1r2 sin 1  2 i 
And the quotient is:
We could factor out
these coefficients

z1  r1
r1
  cos1   2   sin 1   2 i 
z2  r2
r2

Example
Find the product of the complex numbers.
2
2 

z1  2 cos
 i sin

3
3 

11
11 

z2  8 cos
 i sin

6
6 

Try It:
Take a complex number
Multiply it by itself
3(cos   i sin  )
3(cos   i sin  )  3(cos   i sin  )
Square it
Now, multiply this product by the original
Cube it
See a pattern?
9cos 2  i sin 2   3(cos   i sin  )
De Moivre’s Theorem
For a complex number where n is a positive integer:
z  r cos   i sin  
n
n
 r n cos n  i sin n 
Remember where the n ‘s go
Example
Use De Moivre’s to find:

2 3 i

5
Problem Set 6.5.2