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(Section 6.5: Trig and Euler Forms of a Complex Number)
6.33
SECTION 6.5: TRIG (AND EULER / EXPONENTIAL) FORMS
OF A COMPLEX NUMBER
See the
Handout on my website
.
PART A: DIFFERENT FORMS OF A COMPLEX NUMBER
Let
a
,
b
, and
r
be real numbers, and let
θ
be measured in either degrees or radians (in
which case it could be treated as a real number.)
We often let
z
denote a complex number.
Standard or Rectangular Form:
z
=
a
+
bi
We saw this form in
Section 2.4
.
The complex number
a
+
bi
may be graphed in the complex plane as either the
point
a
,
b
( )
or as the position vector
a
,
b
, in which case our analyses from
Section 6.3
become helpful.
Trig Form:
z
=
r
cos
+
i
sin
( )
r
takes on the role of
v
from our discussion of vectors. It is the distance of the
point representing the complex number from 0.
has a role similar to the one it had in our discussion of vectors, namely as a
direction angle, but this time in the complex plane.
This is derived from the Standard Form through the relations:
a
=
r
cos
, and
b
=
r
sin
Recall from
Section 6.3
, with
r
replacing
v
:
r
=
a
2
+
b
2
Choose
such that:
tan
=
b
a
if
a
≠
0
( )
, and
is in the correct Quadrant
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View Full Document(Section 6.5: Trig and Euler Forms of a Complex Number)
6.34
Euler (Exponential) Form:
z
=
re
i
θ
This is useful to derive various formulas in the
Handout
.
In this class, you will not be required to use this form.
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.
 Fall '11
 staff
 Calculus, Real Numbers

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