Polar Form Of Complex Numbers Worksheet With Answer Key - Openstax College Page 12
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25OpenStax-CNX module: m49408
12
try it feature:
Exercise 16
(Solution on p. 18.)
z
= 2 3 (cos (150 ) + isin (150 ))
z
= 2 (cos (30 ) + isin (30 )) .
Find the product and the quotient of
and
1
2
7 Finding Powers of Complex Numbers in Polar Form
De Moivre's Theorem
Finding powers of complex numbers is greatly simpli ed using
. It states that, for a
n
n, z
n
n.
positive integer
is found by raising the modulus to the
th power and multiplying the argument by
It
is the standard method used in modern mathematics.
z = r (cos θ + isin θ)
If
is a complex number, then
a general note label:
n
n
z
= r
[cos (nθ) + isin (nθ)]
(21)
n
n
z
= r
(nθ)
cis
n
where
is a positive integer.
Example 10
Evaluating an Expression Using De Moivre's Theorem
5
(1 + i)
Evaluate the expression
using De Moivre's Theorem.
Solution
Since De Moivre's Theorem applies to complex numbers written in polar form, we must rst
(1 + i)
r.
write
in polar form. Let us nd
2
2
r =
x
+ y
2
2
(22)
r =
(1)
+ (1)
r =
2
y
θ.
tan θ =
Then we nd
Using the formula
x gives
1
tan θ =
1
tan θ = 1
(23)
π
θ =
4
Use De Moivre's Theorem to evaluate the expression.
n
n
(a + bi)
= r
[cos (nθ) + isin (nθ)]
5
5
π
π
(1 + i)
=
2
cos 5
+ isin 5
4
4
5
5π
5π
(1 + i)
= 4 2 cos
+ isin
(24)
4
4
5
2
2
(1 + i)
= 4 2
+ i
2
2
5
(1 + i)
=
4
4i
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