Polar Form Of Complex Numbers Worksheet With Answer Key - Openstax College Page 10
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25OpenStax-CNX module: m49408
10
Example 7
Finding the Rectangular Form of a Complex Number
5
r = 13
tan θ =
.
Find the rectangular form of the complex number given
and
12
Solution
5
y
2
2
2
2
tan θ =
,
tan θ =
,
r =
x
+ y
=
12
+ 5
= 13
If
and
we rst determine
. We then
12
x
y
x
cos θ =
sin θ =
.
nd
r and
r
z = 13 (cos θ + isin θ)
12
5
= 13
+
i
(15)
13
13
= 12 + 5i
12 + 5i.
The rectangular form of the given number in complex form is
try it feature:
Exercise 13
(Solution on p. 18.)
Convert the complex number to rectangular form:
11π
11π
z = 4 cos
+ isin
(16)
6
6
5 Finding Products of Complex Numbers in Polar Form
Now that we can convert complex numbers to polar form we will learn how to perform operations on
complex numbers in polar form.
For the rest of this section, we will work with formulas developed by
de Moivre
French mathematician Abraham
(1667-1754). These formulas have made working with products,
quotients, powers, and roots of complex numbers much simpler than they appear. The rules are based on
multiplying the moduli and adding the arguments.
z
= r
(cos θ
+ isin θ
)
z
= r
(cos θ
+ isin θ
) ,
If
and
then the prod-
a general note label:
1
1
1
1
2
2
2
2
uct of these numbers is given as:
z
z
= r
r
[cos (θ
+ θ
) + isin (θ
+ θ
)]
(17)
1
2
1
2
1
2
1
2
z
z
= r
r
(θ
+ θ
)
2 cis
1
2
1
1
2
Notice that the product calls for multiplying the moduli and adding the angles.
Example 8
Finding the Product of Two Complex Numbers in Polar Form
z
z
,
z
= 4 (cos (80 ) + isin (80 ))
z
= 2 (cos (145 ) + isin (145 )) .
Find the product of
given
and
1
2
1
2
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