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) ▪ Roots of
Complex Numbers
Copyright © 2009 Pearson Addison-Wesley
1.1-3
8.6-3
De Moivre’s Theorem
is a complex number,
then
In compact form, this is written
Copyright © 2009 Pearson Addison-Wesley
1.1-4
8.6-4
Remember the following:
r=
Copyright © 2009 Pearson Addison-Wesley
𝑎2
+ 𝑏 2 , tan
1.1-5
𝑏
θ=
𝑎
8.6-5
Example 1
Find
form.
First write
FINDING A POWER OF A COMPLEX
NUMBER
and express the result in rectangular
in trigonometric form.
Because x and y are both positive, θ is in quadrant I,
so θ = 60°.
Copyright © 2009 Pearson Addison-Wesley
1.1-6
8.6-6
Example 1
FINDING A POWER OF A COMPLEX
NUMBER (continued)
Now apply De Moivre’s theorem.
480° and 120° are coterminal.
Rectangular form
Copyright © 2009 Pearson Addison-Wesley
1.1-7
8.6-7
nth 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
1.1-8
8.6-8
nth Root Theorem
If n is any positive integer, r is a positive
real number, and θ is in degrees, then the
nonzero complex number r(cos θ + i sin θ)
has exactly n distinct nth roots, given by
where
Copyright © 2009 Pearson Addison-Wesley
1.1-9
8.6-9
Note
In the statement of the nth root theorem,
if θ is in radians, then
Copyright © 2009 Pearson Addison-Wesley
1.1-10
8.6-10
Example 2
FINDING COMPLEX ROOTS
Find the two square roots of 4i. Write the roots in
rectangular form.
Write 4i in trigonometric form:
The square roots have absolute value
and argument
Copyright © 2009 Pearson Addison-Wesley
1.1-11
8.6-11
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
1.1-12
8.6-12
Example 2
Copyright © 2009 Pearson Addison-Wesley
FINDING COMPLEX ROOTS (continued)
1.1-13
8.6-13
Example 3
FINDING COMPLEX ROOTS
Find all fourth roots of
rectangular form.
Write
Write the roots in
in trigonometric form:
The fourth roots have absolute value
and argument
Copyright © 2009 Pearson Addison-Wesley
1.1-14
8.6-14
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°.
Copyright © 2009 Pearson Addison-Wesley
1.1-15
8.6-15
Example 3
Copyright © 2009 Pearson Addison-Wesley
FINDING COMPLEX ROOTS (continued)
1.1-16
8.6-16
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.
Copyright © 2009 Pearson Addison-Wesley
1.1-17
8.6-17
Example 4
SOLVING AN EQUATION BY FINDING
COMPLEX ROOTS
Find all complex number solutions of x5 – i = 0. Graph
them as vectors in the complex plane.
There is one real solution, 1, while there are five
complex solutions.
Write 1 in trigonometric form:
Copyright © 2009 Pearson Addison-Wesley
1.1-18
8.6-18
Example 4
SOLVING AN EQUATION BY FINDING
COMPLEX ROOTS (continued)
The fifth roots have absolute value
argument
and
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°}
Copyright © 2009 Pearson Addison-Wesley
1.1-19
8.6-19
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.
Copyright © 2009 Pearson Addison-Wesley
1.1-20
8.6-20
Copyright © 2009 Pearson Addison-Wesley
1.1-21
8.6-21