Are these the only ones that exist for complex numbers?
How are operations like those mentioned above carried out on complex umbers?
Applications and Trigonometric Tools: Real-Life Problem
Learning Objectives
A student will be able to:
Solve everyday problems that require you to use the product and/or quotient theorem of complex numbers in polar form to obtain the correct solution.
Introduction
We have learned how to determine both the product and the quotient of complex numbers that are expressed in polar form. Now it is time to apply these procedures to real – life problems.
1. The electric power (in watts) supplied to an element in a circuit is the product of the voltage e and the current (in amps). Find the expression for the power supplied if and . Note: Use the formula .
Solution:
2. If the angular velocity of a wire rotating through a magnetic field is , the capacitive and inductive reactances are determined by the relation:
and
If , , , and , find the impedance between the current and the voltage.
Solution:
3. In a series alternating current with a resistor, an inductor and a capacitor, , , and . Determine the phase angle.
Solution:
4. For an alternating current circuit in which , , and , find the impedance between the current and the voltage.
Solution: .
De Moivre’s Theorem: Powers and Roots of Complex Numbers
Learning Objectives
A student will be able to:
Use De Moivre’s Theorem to find the powers of complex numbers in polar form.
Introduction
The basic operations of addition, subtraction, multiplication and division of complex numbers have all been explored in this chapter. The addition and subtraction of complex numbers lent themselves best to those in standard form. However multiplication and division were easily performed when the complex numbers were in polar form. Another operation that is performed using the polar form of complex numbers is the process of raising a complex number to a power.
The polar form of a complex number is . If we allow to equal the polar form of a complex number, it is very easy to see the development of a pattern when raising a complex number in polar form to a power. To discover this pattern, it is necessary to perform some basic multiplication of complex numbers in polar form.
If and then:
Likewise, if and then
Again, if and then
De Moivre’s Theorem
These examples suggest a general rule valid for all . We offer this rule and assume its validity for all without formal proof, leaving the proof for later studies. The general rule for raising a complex number in polar form to a power is called De Moivre’s Theorem, and has important applications in engineering, particularly circuit analysis. The rule is as follows:
Let and let be a positive integer.
Notice what this rule looks like geometrically. A complex number taken to the th power has two motions: First, its distance from the origin is taken to the nth power; second, its angle is multiplied by . Conversely, the roots of a number have angles that are evenly spaced about the origin.
Example 1: Find.
Solution:
Using De Moivre’s Theorem:
Example 2: Find
Solution:
Now use De Moivre’s Theorem:
Lesson Summary
In this lesson you discovered the pattern for raising complex numbers in polar form to a power. This pattern was then transferred into a general rule. This general rule is called De Moivre’s Theorem.
Points to Consider
If a complex number in polar formed can be raised to a power, can the roots of a complex number be determined?
If the roots can be determined, will some form of De Moivre’s Theorem be used?
What do powers and roots of complex numbers look like on the complex plane.
Review Questions
Show that , if
Rewrite the following in rectangular form:
Review Answers
Express in polar form:
There are two other cube roots of in the complex plane. Can you find them and plot them on the complex plane? What do the three roots look like geometrically?
nth Root Theorem
Learning Objectives
A student will be able to:
Find the roots of complex numbers in polar form.
Introduction
We have explored all of the basic operations of arithmetic as they apply to complex numbers in standard form and in polar form. The last discovery is that of taking roots of complex numbers in polar form.
We have discovered the general rule for raising a complex number in polar form to a power. This general rule is known as De Moivre’s Theorem. This rule will be used to develop another general rule –one for finding the root of a complex number written in polar form.
As before, let and let the root of be
can be any coterminal angle with .
Therefore, for any integer is an root of if and
The distinct roots of are determined when .
The general rule for finding the roots of a complex number if is:
, where .
Let’s begin with a simple example and we will leave in degrees.
Example 1: Find the two square roots of .
Solution:
Express in polar form.
To find the other root, add to
Example 2: Find the three cube roots of
Solution:
Express in polar form:
Lesson Summary
In this lesson you learned that it was possible to determine the root of a complex number in polar form. De Moivre’s Theorem, which is used to raise a complex number in polar form to a power, can be adapted for finding the roots because roots are merely powers with fractional exponents.
Points to Consider
If the root of a complex number in polar form can be determined, can the solution to an exponential equation be found inn the same way?
What do the roots of a number look like when plotted together on the complex plane?
Review Questions
Find .
Find the principal root of .
Remember the principal root is the positive root i.e. so the principal root is .
Review Answers
Polar From
In standard form and this is the principal root of .
Solve Equations
Learning Objectives
A student will be able to:
Solve equations(find their roots) using the general rule for terms.
Introduction
The roots of a complex number are cyclic in nature. This means that when the roots are plotted on the complex plane, the roots are equally spaced on the circumference of a circle.
Since you began Algebra, solving equations has been an extensive topic. Now we will extend the rules to include complex numbers. The easiest way to explore the process is to actually solve an equation. The solution can be obtained by using De Moivre’s Theorem.
Example 1:
Consider the equation . The solution is the same as the solution of . In other words, we must determine the fifth roots of .
Solution:
Write an expression for determining the fifth roots of
Lesson Summary
In this lesson, you extended your knowledge of De Moivre’s Theorem to include solving equations. The process was the same as that followed to determine the roots of a complex number in polar form.
Points to Consider
If the solutions to the equation were represented graphically, would the result be cyclic in nature?
Does the number of roots have anything to do with the shape of the graph?
Review Questions
Solve the equation
Review Answers
Write an expression for
determining the fourth roots of
Applications, Trigonometric Tools: Powers and Roots of Complex Numbers
Learning Objectives
Incorporate geometry with the results of applying De Moivre’s Theorem.
Introduction
In this lesson we will explore the cyclic nature of the roots of a complex number. The roots of a complex number, when graphed on the complex plane, are equally spaced around a circle. All that is necessary to graph the roots is one of the roots and the radius of the circle.
1. Calculate the three cube roots of and represent them graphically. When you have successfully completed this task, plot the fifth roots of that you found in the previous lesson. What shape did the roots form? A pentagon
Solution:
In standard form, and
The polar form is
The expression for determining the cube roots of is:
For and the three cube roots of are
When these three roots are represented graphically, the three points, on the circle with a radius of , form a triangle. (Three roots resulted in the geometric shape – a triangle)
2. Jessie Neal is an engineer for Eastlink Communications. Her job involves managing the location of antennae and signal towers for mapping relay signals. The figure below shows the location of a transmitting tower and a possible location of antenna for receiving the signal.
The transmitting tower emits the signal that could be picked up by an antenna located at . The location of is not definite and depends upon the strength of the signal between and . The fixed point is units from . and . The length of will determine the location of . As increases, the strength of the signal decreases. By using the Pythagorean theorem and polar coordinates, Josie is able to determine the length of and thus interpret the strength of the signal.
Calculate if , , and .
Solution: .
Trigonometric Applications:
The computer software program, Autograph, is an excellent resource for graphing the roots of a complex number in polar form. The coordinates are easily entered and the software plots the points when the coordinates are entered. This program also allows you to edit the axis so that the resulting graph fits nicely into a document.
TABLE OF CONTENTS
CK-12 License
Chapter 1: Trigonometry and Right Angles
Chapter 2: Circular Functions
Chapter 3: Trigonometric Identities
Chapter 4: Inverse Functions and Trigonometric Equations
Chapter 5: Triangles and Vectors
Chapter 6: Polar Equations and Complex Numbers.
CK-12 Trigonometry
CK-12 Trigonometry Page 32