Transcript Chapter 8 Section 6
8
Applications of Trigonometry
Copyright © 2009 Pearson Addison-Wesley
8.6-1
8
Applications of Trigonometry
8.1
The Law of Sines
8.2
The Law of Cosines
8.3
Vectors, Operations, and the Dot Product
8.4
Applications of Vectors
8.5
Trigonometric (Polar) Form of Complex Numbers; Products and Quotients
8.6
De Moivre’s Theorem; Powers and Roots of Complex Numbers
8.7
Polar Equations and Graphs
8.8
Parametric Equations, Graphs, and Applications Copyright © 2009 Pearson Addison-Wesley
8.6-2
8.6
De Moivre’s Theorem; Powers and Roots of Complex Numbers
Powers of Complex Numbers (De Moivre’s Theorem) ▪ Complex Numbers Roots of
8.6-3
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1.1-3
De Moivre’s Theorem is a complex number, then In compact form, this is written Copyright © 2009 Pearson Addison-Wesley
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Remember the following: Copyright © 2009 Pearson Addison-Wesley r = 𝑎 2 + 𝑏 2 , tan θ = 𝑏 𝑎
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Example 1
FINDING A POWER OF A COMPLEX NUMBER
Find and express the result in rectangular form.
First write in trigonometric form.
Because
x
so
θ
= 60 °.
and
y
are both positive,
θ
is in quadrant I,
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Example 1
FINDING A POWER OF A COMPLEX NUMBER (continued)
Now apply De Moivre’s theorem.
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480 ° and 120° are coterminal.
Rectangular form
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n
th Root For a positive integer
n
, the complex number
a
+
bi
is an
nth root
of the complex number
x
+
yi
if Copyright © 2009 Pearson Addison-Wesley
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n
th Root Theorem If
n
is any positive integer,
r
real number, and
θ
is a positive is in degrees, then the nonzero complex number
r
(cos
θ
+
i
sin
θ
) has exactly
n
distinct
n
th roots, given by where Copyright © 2009 Pearson Addison-Wesley
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Note
In the statement of the
n
th root theorem, if
θ
is in radians, then Copyright © 2009 Pearson Addison-Wesley
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Example 2
FINDING COMPLEX ROOTS
Find the two square roots of 4
i
. Write the roots in rectangular form.
Write 4
i
in trigonometric form: The square roots have absolute value and argument
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Example 2
FINDING COMPLEX ROOTS (continued)
Since there are two square roots, let
k
= 0 and 1.
Using these values for , the square roots are Copyright © 2009 Pearson Addison-Wesley
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Example 2
FINDING COMPLEX ROOTS (continued)
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Example 3
FINDING COMPLEX ROOTS
Find all fourth roots of Write the roots in rectangular form.
Write in trigonometric form: The fourth roots have absolute value and argument
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Example 3
FINDING COMPLEX ROOTS (continued)
Since there are four roots, let
k
= 0, 1, 2, and 3.
Using these values for
α
, the fourth roots are 2 cis 30 °, 2 cis 120°, 2 cis 210°, and 2 cis 300°.
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Example 3
FINDING COMPLEX ROOTS (continued)
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Example 3
FINDING COMPLEX ROOTS (continued)
The graphs of the roots lie on a circle with center at the origin and radius 2. The roots are equally spaced about the circle, 90 ° apart.
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Example 4
SOLVING AN EQUATION BY FINDING COMPLEX ROOTS
Find all complex number solutions of
x
5 – them as vectors in the complex plane.
i
= 0. Graph There is one real solution, 1, while there are five complex solutions.
Write 1 in trigonometric form:
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Example 4
SOLVING AN EQUATION BY FINDING COMPLEX ROOTS (continued)
The fifth roots have absolute value and argument Since there are five roots, let
k
= 0, 1, 2, 3, and 4.
Solution set: {cis 0 °, cis 72°, cis 144°, cis 216°, cis 288°}
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Example 4
SOLVING AN EQUATION BY FINDING COMPLEX ROOTS (continued)
The graphs of the roots lie on a unit circle. The roots are equally spaced about the circle, 72 ° apart.
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