Transcript 6.5 Trigonometric Form of a Complex Number

Digital Lesson
Trigonometric Form of
a Complex Number
In the complex plane, every complex number
corresponds to a point.
Example:
Plot the points 3 + 4i and
–2 – 2i in the complex plane.
Imaginary axis
4
(3, 4)
or
3 + 4i
2
Real
axis
–2
(– 2, – 2)
or
– 2 – 2i
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
2
–2
2
The absolute value of the complex number z = a + bi is
the distance between the origin (0, 0) and the point (a, b).
| a  bi |  a  b
2
2
Example:
Plot z = 3 + 6i and find its absolute value.
Imaginary axis
z  3 6
2
8
6
z = 3 + 6i
 9  36
4
–4
2
3 5 units
–2
4
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
2
Real
axis
 45
3 5
3
To write a complex number a + bi in trigonometric
form, let  be the angle from the positive real axis
(measured counter clockwise) to the line segment
connecting the origin to the point (a, b).
Imaginary axis
a = r cos 
b = r sin 
(a, b)
r  a 2  b2
r
b
a  bi  (r cos  )  (r sin  )i

a
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Real
axis
4
The trigonometric form of a complex number
z = a + bi is given by
z = r(cos  + i sin )
where a = r cos , b = r sin , r  a 2  b2 , and tan   b .
a
The number r is the modulus of z, and  is the
argument of z.
Example:

z  1 cos   i sin 
2
3
3
modulus
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

argument
5
Example:
Write the complex number z = –7 + 4i in trigonometric
form.
z  r  7  4i
Imaginary axis
 (7) 2  42  65
z = –7 + 4i
z  65
tan   b   4
a
7
tan 1  4  29.74
7
  180  29.74  150.26
 
150.26°
Real
axis
z  r  cos θ  i sin θ 
 65(cos150.26  i sin150.26)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
6
Example:
Write the complex number 3.75 cos 3  i sin 3
4
4
in standard form a + bi.


cos 3   2
4
2
sin 3  2
4
2


2
2
z  3.75  

i
2 
 2
 15 2  15 2 i Standard form
8
8
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
7
Example:
Write the complex number 3  cos 330  i sin 330 
2
in standard form a + bi.
cos 330  3
2
sin 330   1
2
 3 1 
3
z 
 i
2 2 2 
3
3

 3 i Standard form
4
4
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
8
Graphing Utility:
Write the complex number 3.75 cos 3  i sin 3
4
4
in standard form a + bi.


[2nd] [decimal point]

3.75 cos 3  i sin 3
4
4

 2.652  2.652i
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
9