Download Trigonometric Form of Complex Numbers

Trigonometric
Form of Complex
Numbers 6.6a
The first stuff in our last section
of the chapter!
But first, remind me – what’s a
complex number???
A complex number is one that can be written in the form
a  bi
where a and b are real numbers. The real number a is the
real part, the real number b is the imaginary part, and
a + bi is the standard form.
And of course, remember the definition of the imaginary number:
i  1
In Sec. 6.1, we learned how to write a vector in
“trigonometric form”:
v  ai  bj
v   v cos θ  i   v sin θ  j
 v  cosθi  sin θj
Now, we will do something similar
with complex numbers…
Recall how we graph
complex numbers:
Imaginary
Axis
P(a, b)
z = a + bi
r
b
0
a
a  r cosθ b  r sin θ
z  a  bi
  r cosθ    r sin θ  i
Real
Axis
r  z  a b
2
2
 r  cosθ  i sin θ 
b
tan θ 
a
Definition: Trigonometric Form
of a Complex Number
The trigonometric form of the complex number z = a + bi is
z  r  cosθ  i sin θ 
The number r is the absolute value or modulus of z,
and 0 is an argument of z.
 Is the argument of any particular complex number unique?
Practice changing forms of
complex numbers
Switch forms of the given complex number, for
0  θ  2π
(between trigonometric form and standard form)
1  3i
How about a graph???
r  1  3i 
1
Reference angle:
π

3
2

 3
so…
2
2
 π  5π
θ  2π     
 3 3
5π
5π
1  3i  2 cos  2i sin
3
3
Practice changing forms of
complex numbers
Switch forms of the given complex number, for
0  θ  2π
3  4i
3  4i  5  cos 4.069  i sin 4.069
Practice changing forms of
complex numbers
Switch forms of the given complex number, for
0  θ  2π
π
π

3  cos  i sin 
6
6

In this case, simply evaluate the trigonometric functions…
 3 1  3 3 3
π
π

3  cos  i sin   3 
 i  
 i
6
6
2
2

 2 2 
Practice changing forms of
complex numbers
Switch forms of the given complex number, for
0  θ  2π
17  cos105  i sin105 
17  cos105  i sin105  1.067  3.983i
Practice changing forms of
complex numbers
Switch forms of the given complex number, for
5i
π
π

5i  5  cos  i sin 
2
2

0  θ  2π
Whiteboard Problems:
Switch forms of the given complex number, for
1. 3  3i
2. 8(cos 210  i sin 210 )
3.
5(cos


 i sin )
4
4
0  θ  2π