When you worked with a system of linear equations with two unknowns, finding the point of intersection of the equations meant finding the coordinates of the point that satisfied both equations. If the equations are rectangular equations for curves, determining the point(s) of intersection of the curves involves solving the equations algebraically since each point will have one ordered pair of coordinates associated with it.
Example 1: Solve the following system of equations algebraically:
Solution:
Before solving the system, graph the equations to determine the number of points of intersection.
The graph of is an ellipse and that of is a parabola. There are three points of intersection.
The estimated points of intersection are labeled on the graph. To determine the exact values of these points, algebra must be used.
Using the quadratic formula,
These values must be substituted into one of the original equations.
The three points of intersection as determined algebraically in Cartesian representation are , and .
The points of intersection are those shown above. However, if we are working with polar equations to determine the polar coordinates of a point of intersection, we must remember that there are many polar coordinates that represent the same point. Remember that switching to polar form changes a great deal more than the notation. Unlike the Cartesian system which has one name for each point, the polar system has an infinite number of names for each point. One option would be to convert the polar coordinates to rectangular form and then to convert the coordinates for the intersection points back to polar form. Perhaps the best option would be to explore some examples. As these examples are presented, be sure to use your graphing calculator to create your own visual representations of the equations presented. To view the intersection points, use the zoom function and the trace function on the calculator.
Example 2: Determine the polar coordinates for the intersection point(s) of the following polar equations: and
Solution:
Begin with the graph. Using the process described in the technology segment of section one in this chapter; create the graph of these polar equations on your graphing calculator. Once the graphs are on the screen, use the trace function and the arrow keys to move the cursor around each graph. As the cursor is moved, you will notice that the equation of the curve is shown in the upper left corner and the values of , are shown (in decimal form) at the bottom of the screen. The values change as the cursor is moved.
in the first quadrant and in the fourth quadrant.
The obvious points of intersection are and . However, these two solutions only cover the possible values . If you consider that is true for an infinite number of theta these solutions must be extended to include and . Now the solutions include all possible rotations.
This example was solved as any system of rectangular equations would be solved. Does this approach work all the time?
Example 3: Find the intersection of the graphs of and
Solution:
Begin with the graph. You can create these graphs using your graphing calculator.
If you consider that is true for an infinite number of theta as was in the previous example, the same consideration must be applied to include all possible solutions. To prove if the origin is indeed an intersection point, we must determine whether or not both curves pass through .
From this investigation, the point was on the curve and the point was on the curve . Because the second coordinates are different, it seems that they are two different points. However, the coordinates represent the same point . The intersection points are and
Sometimes it is helpful to convert the equations to rectangular form, solve the system and then convert the polar coordinates back to polar form.
Example 4: Find the intersection of the graphs of and
Solution:
Begin with the graph:
The equations are now in rectangular form. Solve the system of equations.
Substituting these values into the first equation:
The points of intersection are and
The rectangular coordinates are and . Converting these coordinates to polar coordinates give the same coordinates in polar form. The points can be converted by using the angle menu of the TI calculator. This process was shown in the previous lesson.
We will now return to Josie and try to solve her problem. One mural is represented by the equation and the other by . To determine where they will intersect, we will begin with a graph.
Josie’s murals would intersect and two points and
Lesson Summary
In this lesson you learned how to graph polar equations to see the points of intersections of the polar curves. In addition to seeing the points, you learned how to determine the coordinates of these intersection points using several approaches. The fact that many polar coordinates can represent the same point was revisited as well.
Points to Consider
Will polar curves always intersect?
If not, when will intersection not occur?
If two polar curves have different equations, can they be the same curve?
Review Questions
Find the intersection of the graphs of and
Find the intersection of the graphs of and
Review Answers
There appears to be one point of intersection.
The point of intersection is
The coordinates represent the same point
.
Equivalent Polar Curves
Learning Objectives
A student will be able to:
Graph equivalent polar curves.
Recognize equivalent polar curves from their equations.
Understand that equivalent polar curves are often symmetrical about different axis but are still equal.
Understand why equivalent polar curves do not intersect.
Introduction
The expression “same only different” comes into play in this lesson. We will graph two distinct polar equations that will produce two equivalent graphs. Use your graphing calculator and create these curves as the equations are presented.
Previously, graphs were generated of a limaçon, a dimpled limaçon, a looped limaçon and a cardioid. All of these were of the form or . The easiest way to see what polar equations produce equivalent curves is to use either a graphing calculator or a software program like Geometer’s Sketchpad to generate the graphs of various polar equations.
Example 1: Plot the following polar equations and compare the graphs.
a)
b)
Solution:
These graphs represent
Although the polar equations are different
the resulting graphs shows that they are equivalent.
These graphs represent the equations
The difference between these equations is the values for theta. Now it is visible that the equations are equal.
Example 2:
Graph the equations
Describe the graphs.
Solution:
Both equations, one in rectangular form and one in polar form, are circles with a radius of and center at the origin.
Example 3:
Graph the equations
Describe the graphs.
There is not a visual representation shown here, but on your calculator you should see that the graphs are circles centered at with a radius .
Lesson Summary
In this lesson you were introduced to the notion that the graphs of solution sets of polar curves can be equivalent. It is difficult to predict equivalent graphs by looking at the equation in isolation. However, once the graphs are created, the equivalence of the sets is visible.
Points to Consider
When looking for intersections, which representation is easier to work with? Look over the examples and find some in which doing the algebra in polar coordinates is more direct that finding intersections in Cartesian form.
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Review Questions
Write the rectangular equation in polar form and graph the equations.
Graph the equations Are they equivalent?
Review Answers
Both equations produced a circle with center
and a radius of
.
Yes, the equations produced the same graph so they are equivalent.
Applications, Technological Tools: Systems of Polar Equations
Learning Objectives
A student will be able to:
Understand the useful application of the intersection of polar curves as it applies to real world problems.
Introduction
In this section we will look at some real world applications of the topics visited in this lesson.
Stephanie is making a quilt. In each block, she is sewing a rose with petals and adding a sheer, metallic overlay on top of the rose. She plans to repeat this pattern in every fourth block of her quilt. To keep the pattern repeating in a perfect manner, Stephanie must decide the exact position of the overlay on the rose. If she knows this, she can be certain that every fourth block will repeat exactly. The limaçon, which is the shape of the overlay, was designed by using the equation , while the shape of the rose was designed by using the equation . Create a graphic representation of this design so you can explain the intersection points to Stephanie.
There appear to be intersection points between the limaçon, and the rose. However, the true points of intersection are the two points in the first quadrant and the two points in the third quadrant. At the other four intersection points, the values on the rose are negative.
Technology Application: Using the TI-8s calculator to graph polar curves is an excellent learning tool. You can actually simulate the graphing process by using the simulation mode.
On the [MODE] menu of the calculator, scroll down to Radian Degree and highlight Degree. Continue to the next line, and highlight Pol. Continue to scroll down and highlight Simul. Press [ZOOM] to access a square viewing window.
Press and type in and in . Now press [GRAPH].
You will see the graph plot slowly. To ensure that you see the entire graphing process, press [WINDOW] and enter step as a small number. The smaller the number, the slower it graphs. The graph pauses at various intervals throughout the graphing process. These points can be determined by using the trace feature. As the graph is traced the various values appear on the screen.
You can see the graph in this screen capture of the calculator.
Vocabulary
Polar Coordinates
The polar coordinates of a point are written in the form , where is the distance from the pole to point and is the measure of an angle between and the polar axis (which aligns with the the positive axis.)
Polar Equation
An equation which uses polar coordinates.
Polar Graph
A graph that represents the set of all points which satisfy a given polar equation.
Recognize
Recognize .
Learning Objectives
A student will be able to:
Understand the concept of a complex number.
Recognize a complex number.
Introduction
In solving algebraic equations, you have probably come across equations such as that have no solutions because no real number squared equals a negative number. While using the quadratic formula, you have probably encountered a similar problem, when produces a negative value and there is no real solution. Complex numbers are introduced to produce solutions to these equations. Even though these numbers don’t exist on the real number line, they follow strict arithmetic laws similar to the real numbers, and it is convenient to have a larger system where all algebraic equations have solutions.
The square of any positive number or any negative number results in a positive number. Therefore, it seems natural to say that it is impossible to square any real number and have the result be a negative number. In order to include square roots of negative numbers, we must define a new number system. These numbers, called the complex numbers, are a formal extension of the real numbers. It might seem arbitrary or capricious to define a number that is “imaginary” and does not exist in the sense that counting numbers do, but the complex system has remarkable mathematical properties and applies in a surprising number of real-world instances. The first important insight was the Fundamental Theorem of Algebra, proved by Gauss at age 21. All equations over the complex numbers have solutions. More specifically, all polynomials of degree with real coefficients have roots in the complex system. So all quadratics have two roots; all cubics have three etc.
To build the complex number system, we begin with the simplest root of a negative number: . The symbol is defined as the imaginary unit and is represented by the symbol . The only thing we know about is what we know about the square root of any number—that when you multiply it by itself it equals the number inside. As a result, if then . We also extend the well-known rule for square roots of positive numbers, , to square roots of negative numbers. The rule holds when or is negative, but not both, as we will see below. First, here are some applications of this extended rule.
Example 1: Express the following square roots in terms of .
a)
b)
c)
Solution:
a)
b)
c)
Writing the solution to , with in front of the radical, shows that is not under the radical sign with .
Operations with radicals are defined under the assumption that all letters represent positive numbers. For example is valid if neither nor is negative.
The radical expression can be written as but not as since this later representation will produce an incorrect solution of . The correct solution is
Example 2: Simplify the following expression:
a)
b)
c)
Solution:
a)
b)
c)
Lesson Summary
In this lesson you learned to determine the square root of a negative number. You also learned that operations performed on radicals do not apply to negative radicands. However, you did learn to apply the rule for the product of radicals to reflect the product of square roots of negative numbers.
Points to Consider
Can complex numbers exist in another form?
Can complex numbers be expressed in rectangular form? In polar form?
Do complex numbers fit in the Real Number System?
Review Questions
Express each of the following in terms of . Write each solution in simplest form.
Review Answers
Standard Form of Complex Numbers (a + bi)
Learning Objectives
A student will be able to:
Recognize the standard form of a complex number.
Understand the term imaginary as it applies to complex numbers.
Write complex numbers in standard form.
Introduction
You are now able to recognize a complex number as defined in the previous lesson. You are also able to express them in terms of . In this lesson you will learn to express complex numbers in rectangular/standard form. It is in this form that we will later learn to perform basic operations with complex numbers.
Using real numbers and the imaginary unit , a new kind of number can be defined. A complex number is any number that can be written in the form , where and are real numbers. If and , the number is in the form , which is referred to as a pure imaginary number. If , then is a real number. The form is known as the rectangular form of a complex number. In the rectangular form, is called the real part and is the imaginary part. As a result, the complex numbers include both the real numbers and the pure imaginary numbers.
Although we think of the word imaginary as portraying something that does not exist, such is not the case with respect to complex numbers. They are as real as real numbers in the
sense that they are well-defined concepts (neither real number nor imaginary numbers exist in a physical sense!) As well, the term complex indicates complicated and again this is not the case with complex numbers. The rules are quite simple. Before we move on to basic operations with complex numbers, we must first explore the notion of equality of complex numbers.
From its definition, a complex number is the sum of a real number and an imaginary number.
Since the sum is one of two distinct parts, the number is not negative or positive as we would normally think of these values. Instead, each real part and each imaginary part are positive or negative.
CK-12 Trigonometry Page 29
