Polar Form Of Complex Numbers Worksheet With Answer Key - Openstax College Page 14
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25OpenStax-CNX module: m49408
14
try it feature:
Exercise 19
(Solution on p. 18.)
16 (cos (120 ) + isin (120 )) .
Find the four fourth roots of
Access these online resources for additional instruction and practice with
media feature label:
polar forms of complex numbers.
1
The Product and Quotient of Complex Numbers in Trigonometric Form
2
De Moivre's Theorem
9 Key Concepts
a + bi
Complex numbers in the form
are plotted in the complex plane similar to the way rectangular
coordinates are plotted in the rectangular plane. Label the x-axis as the real axis and the y-axis as
the imaginary axis. See Example 1.
The absolute value of a complex number is the same as its magnitude.
It is the distance from the
2
2
z =
a
+ b
.
origin to the point:
See Example 2 and Example 3.
2
2
x = rcos θ, y = rsin θ,
r =
x
+ y
.
z =
To write complex numbers in polar form, we use the formulas
and
Then,
r (cos θ + isin θ) .
See Example 4 and Example 5.
To convert from polar form to rectangular form, rst evaluate the trigonometric functions.
Then,
r.
multiply through by
See Example 6 and Example 7.
To nd the product of two complex numbers, multiply the two moduli and add the two angles. Evaluate
the trigonometric functions, and multiply using the distributive property. See Example 8.
To nd the quotient of two complex numbers in polar form, nd the quotient of the two moduli and
the dierence of the two angles. See Example 9.
n
z
,
r
n,
θ
n.
To nd the power of a complex number
raise
to the power
and multiply
by
See Example 10.
Finding the roots of a complex number is the same as raising a complex number to a power, but using
a rational exponent. See Example 11.
10 Section Exercises
10.1 Verbal
Exercise 20
(Solution on p. 18.)
a + bi.
A complex number is
Explain each part.
Exercise 21
What does the absolute value of a complex number represent?
Exercise 22
(Solution on p. 18.)
How is a complex number converted to polar form?
Exercise 23
How do we nd the product of two complex numbers?
Exercise 24
(Solution on p. 19.)
What is De Moivre's Theorem and what is it used for?
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