Polar Form Of Complex Numbers Worksheet With Answer Key - Openstax College Page 11
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25OpenStax-CNX module: m49408
11
Solution
Follow the formula
z
z
= 4 2 [cos (80 + 145 ) + isin (80 + 145 )]
1
2
z
z
= 8 [cos (225 ) + isin (225 )]
1
2
5π
5π
z
z
= 8 cos
+ isin
(18)
1
2
4
4
2
2
z
z
= 8
+ i
1
2
2
2
z
z
=
4 2
4i 2
1
2
6 Finding Quotients of Complex Numbers in Polar Form
The quotient of two complex numbers in polar form is the quotient of the two moduli and the dierence of
the two arguments.
z
= r
(cos θ
+ isin θ
)
z
= r
(cos θ
+ isin θ
) ,
If
and
then the quo-
a general note label:
1
1
1
1
2
2
2
2
tient of these numbers is
z
r
=
[cos (θ
θ
) + isin (θ
θ
)] , z
= 0
1
1
1
2
1
2
2
z
r
2
2
(19)
z
r
=
(θ
θ
) , z
= 0
1
1
cis
1
2
2
z
r
2
2
Notice that the moduli are divided, and the angles are subtracted.
Given two complex numbers in polar form, nd the quotient.
how to feature:
r
.
1
1.Divide
r
2
θ
θ
.
2.Find
1
2
r
z = r (cos θ + isin θ) .
r
1
,
θ
θ
3.Substitute the results into the formula:
Replace
with
and replace
with
1
r
2
θ
.
2
r.
4.Calculate the new trigonometric expressions and multiply through by
Example 9
Finding the Quotient of Two Complex Numbers
z
= 2 (cos (213 ) + isin (213 ))
z
= 4 (cos (33 ) + isin (33 )) .
Find the quotient of
and
1
2
Solution
Using the formula, we have
z
2
=
[cos (213
33 ) + isin (213
33 )]
1
z
4
2
z
1
=
[cos (180 ) + isin (180 )]
1
z
2
2
z
1
=
[ 1 + 0i]
(20)
1
z
2
2
z
1
=
+ 0i
1
z
2
2
z
1
=
1
z
2
2
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