CK-12 Trigonometry
Page 27
a. Create a polar graph of this area?
b. If all the seats are occupied and each seat takes up of space, how many people will be seated in the bleachers?
Solution:
Now that the region has been graphed, the next step is to calculate the area of this sector. To do this, use the formula .
The number of people in the bleachers is .
Example 3: When Valentine’s Day arrives, hearts can be seen everywhere. As an alternative to purchasing a greeting card, use a computer to create a heart shape. Write an equation that could be used to create this heart and be careful to ensure that it is displayed in the correct position.
Solution:
The classical curve that resembles a heart is a cordioid. You may have to experiment with the equation to create a heart shape that is displayed in the correct direction.
One example of an equation that produces a proper heart shape is .
You can create other hearts by replacing the number in the equation. Another equation is
Example 4: For centuries, people have been making quilts. These are frequently created by sewing a uniform fabric pattern onto designated locations on the quilt. Using the equation that models a rose curve, create three patterns that could be used for a quilt. Write the equation for each rose and sketch its graph. Explain why the patterns have different numbers of petals. Can you create a sample quilt?
Solution:
a.
b.
c.
The rose curve is a graph of the polar equation of the form or .
If is odd, then the number of petals will be equal to . If is even, then the number of petals will be equal to 2 .
A Sample Quilt:
Graphs of Polar Equations
Learning Objectives
A student will be able to:
Use the TI graphing calculator to create the graphs of polar equations.
Introduction
In today’s world mathematics is not always done using pencil and paper. In a world of technology, graphs can be created very quickly by using a graphing calculator or a computer program. Both are capable of performing mathematical computations accurately and quickly. Since calculators have become an essential item for all students of mathematics, we will focus on using the TI calculator to create graphs of polar equations.
You have all become familiar with the graphs of polar equations. Now you will use technology, the TI graphing calculator, to create these graphs. The TI-83, TI-83 Plus and the TI-84 are very popular graphing calculators used by math students. However, there are steps that must be followed in order to graph polar equations correctly on the graphing calculator. We will go through the step by step process to plot the polar equation .
Example 1: Graph using the TI-83 graphing calculator.
Press the [MODE] button. Scroll down to Func and over to highlight Pol. Also, while on this screen, make sure that Radian is highlighted. Now you must edit the axes for the graph. Press [WINDOW] [ENTER] [ENTER] [ENTER] [ENTER] [ENTER] [ENTER] [ENTER] [ENTER] [ENTER]. When you have completed these steps, the screen should look like this:
The second WINDOW shows part of the first screen since you had to scroll down to access the remaining items.
Enter the equation. Press Press [GRAPH].
Sometimes the polar equation you graph will look more like an ellipse than a circle. If this happens, press [ZOOM] to set a square viewing window. This will make the graph appear like a circle.
Polar to Rectangular
Learning Objectives
A student will be able to:
Convert from polar to rectangular coordinates.
Write an equation given in polar form in rectangular form.
Introduction
Look at the following diagrams. What do you think would happen if we replaced the rectangular coordinate system, which measures how far a point is from the and axes, by a new coordinate system, which instead measures how far a point is from the origin and what angle it has with respect to a ray called the polar axis (which is aligned with the positive axis.) This new system is called a polar coordinate system, as it is focused around a central pole, or point. Figure 1 below shows a rectangular coordinate system and Figure 2 shows a polar coordinate system.
Polar to Rectangular
Just as and are usually used to designate the rectangular coordinates of a point, and (the Greek letter theta) are usually used to designate the polar coordinates of the point. is the distance of the point to the origin. is the angle that the line from the origin to the point makes with the positive axis. The diagram below shows both polar and Cartesian coordinates applied to a point . The pole is the origin and the polar axis is the positive side of the axis. By applying trigonometry, we can obtain equations that will show the relationship between polar coordinates and the rectangular coordinates
The point has the polar coordinates and the rectangular coordinates .
Therefore
These equations, also known as coordinate conversion equations will enable you to convert from polar to rectangular form.
Example 1: Given the following polar coordinates, find the corresponding rectangular coordinates of the points:
Solution:
a)
For and
b)
For and
The rectangular coordinates of W are approximately
The rectangular coordinates of H are or approximately
In addition to writing polar coordinates in rectangular form, the coordinate conversion equations can also be used to write polar equations in rectangular form.
Example 2: Write the polar equation in rectangular form.
Solution:
The equation is now in rectangular form. The and have been replaced. However, the equation, as it appears, does not model any shape with which we are familiar. Therefore, we must continue with the conversion.
The rectangular form of the polar equation represents a circle with its centre at and a radius of .
This is the graph represented by the polar equation for or the rectangular form .
Example 3: Write the polar equation in rectangular form and graph the result.
Solution:
The graph of is a horizontal line passing through and parallel to the axis. .
Lesson Summary
In this lesson we have learned how to convert polar coordinates and polar equations to rectangular form. This has been accomplished by using the coordinate conversion equations. We will use a similar format to in the next lesson to convert from rectangular form to polar form. Each coordinate system has its benefits and drawbacks. Tasks that are simple in one system may be very complicated in another. For example, the equation for a line is simple in
Points to Consider
When we convert coordinates from polar form to rectangular form, the process is very straightforward. However, when converting a coordinate from rectangular form to polar form there are some choices to make. For example the point 1, could translate to , or to or to , and so on.
How does your graphing calculator confront the above problem when converting a rectangular coordinate to a polar coordinate?
How many solutions should you provide when doing these conversions?
How is converting from polar form to rectangular form and vice versa different?
Review Questions
For the following polar coordinates that are shown on the graph, determine the rectangular coordinates for each point.
Write the polar equation in rectangular form and define the graph.
Review Answers
For and
Rectangular to Polar
Learning Objectives
A student will be able to:
Convert rectangular coordinates to polar coordinates.
Convert equations given in rectangular form to equations in polar form.
Introduction
After having a hip replacement, the doctor will order the patient not to bend over for a perio
d of six weeks. To retrieve fall en objects, canes are equipped with a "hand" at the end of a detachable arm. The hand acts as a grabber and can be manipulated by the user to pick up objects. If the hand is to move from a point with rectangular coordinates of to another point with rectangular coordinates , what polar equation can be used to represent this straight line movement? We will address this problem later in the lesson after we learn to convert from rectangular from to polar form.
Rectangular to Polar
When converting rectangular coordinates to polar coordinates, we must remember that there are many possible polar coordinates. We will agree that when converting from rectangular coordinates to polar coordinates, one set of polar coordinates will be sufficient for each set of rectangular coordinates. Most graphing calculators are programmed to complete the conversions and they too, provide one set of coordinates for each conversion. The conversion of rectangular coordinates to polar coordinates is done using the Pythagorean Theorem and the Arctangent function. The Arctangent function only calculates angles in the first and fourth quadrants so radians must be added to the value of for all points with rectangular coordinates in the second and third quadrants.
In addition to these formulas, is also used in converting rectangular coordinates to polar form.
Example 1: Convert the following rectangular coordinates to polar form.
and
For and The point is located in the fourth quadrant and .
The polar coordinates of are
For and . The point is located in the third quadrant and .
The polar coordinates of are
To write a rectangular equation in polar form, the conversion equations of and are used.
Example 2: Write the rectangular equation in polar form. Remember if then and .
Example 3: Write the rectangular equation in polar form. Remember and .
If the graph of the polar equation is the same as the graph of the rectangular equation, then the conversion has been determined correctly.
The rectangular equation represents a circle with center and a radius of .
The polar equation is a circle with center and a radius of .
We will now return to the problem involving the grabber and the cane. The two points were given by the rectangular coordinates and . The equation of the straight line that passes through these points is . To express this equation in polar form, remember .
Lesson Summary
In this lesson we learned how to convert from rectangular form to polar form for both coordinates and equations. When doing these operations, the conversion equations were different than those used in the previous lesson. Although there are many possible solutions when converting rectangular coordinates to polar coordinates, they all represent the same point.
Points to Consider
Are there any advantages to using polar coordinates instead of rectangular coordinates? List any situations in which this is the case. What types of curves are easier to draw with polar coordinates?
List situations in which rectangular coordinates are preferable.
Will polar coordinates be useful in graphing polar curves?
Can graphing two different polar equations ever produce the equivalent curves? Can this ever be true of rectangular equations?
Review Questions
Write the following rectangular points in polar form.
and
Write the rectangular equation in polar form and sketch the graph.
Review Answers
For and . The point is located in the second quadrant and .
The polar coordinates for the rectangular coordinates are
For and . The point is located in the fourth quadrant and .
The polar coordinates for the rectangular coordinates are
The graph of contains the single point, the origin, produced by the graph of . Therefore the polar form of the equation is the single equation:
Polar Equations and Complex Numbers
Conic Section Transformations
Learning Objectives
A student will be able to:
Recognize the curves that are collectively known as conics.
Understand the terms focus, directrix and focal axis as they apply to conics.
Write the equation of a parabola and an ellipse in standard form.
Write polar equations of conics.
Recognize transformations of polar curves and change the equations to produce these transformations.
Introduction
Are you in the dark? Are you prepared for the next power outage in your neighborhood? If you are not totally prepared, at least make sure that you have a dependable flashlight nearby- one that emits a good light and of course has functioning batteries installed. A flashlight is a unique device that uses the characteristics of a parabola in its structure. If you consider the focus as the location of the filament (light source) the light rays are emitted as lines parallel to the axis of symmetry. To produce the best rays, the filament must be placed in the proper spot. If the mirror of the flashlight has a diameter of . and a depth of , how far from the vertex should the filament of the light bulb be placed to yield the best parallel rays of emission?
On a hot summer day, many of us enjoy the refreshing taste of an ice cream scooped into a sugar cone. The cone is hollow and symmetrical about an imaginary line, the axis, which extends through the center of the cone perpendicular to the base. A conic section is simply a thin section of the cone. To better understand these sections of a cone, two cones are lined up vertically tip to tip. Cones aligned in this manner form a circular conical surface. If you look at the figure below, the cones are being sliced by a plane. The manner in which the plane intersects the cones determines the shape of the section. Below is a view of the three standard types of conic sections:
An ellipse is the result of the intersection of a cone on both sides by a plane that is not parallel to the circular base.
A parabola is the result of the intersection of a cone on one side by a plane that is not parallel to the circular base.
A hyperbola is the result of the intersection of a cone and a plane perpendicular to the circular base.
All of these conics have standard equations that are shown in the table below:
Ellipse
or if the center is
or if the center is translated to
Parabola
if the parabola opens up or down. if the parabola opens right or left
if the vertex is translated to and the parabola opens up or down.
if the vertex is translated to and the parabola opens right or left.
Hyperbola
or if the center is
if the center is translated to
What types of transformations can be performed on these conics?
Parabola – The standard parabola can be reflected vertically across the axis. The vertex can be translated from the origin horizontally and/or vertically. The parabola can also be stretched vertically. These transformations can be seen best when the equation is written in this form: , where is the horizontal translation, is the vertical translation, and is the vertical stretch.
Ellipse – For an ellipse of the above form, there are two lines of symmetry, one horizontal and one vertical. The center of the ellipse can be changed by translating the ellipse horizontally and/or vertically.
Hyperbola – The branches of the hyperbola can extend right and left if the foci are on the axis or up and down if the foci are on the axis. The center can also be translated horizontally and/or vertically.
It is time to investigate graphs of the equations of these conics in order to obtain their standard equations. We will begin with the parabola and move on to the ellipse.
A parabola can be defined as the set of all points in a plane that are equidistant from a fixed line (the directrix) and a fixed point (the focus) in the plane.
The axis is the line of symmetry.
The vertex is the point where the parabola
intersects the axis and it is midway between the focus and the directrix.
The focal axis is the perpendicular line passing through the focus to the directrix.
To obtain the standard form of the equation for this parabola, we will use the focus and the directrix . We must prove that a point on the parabola equidistant from the focus and the directrix satisfies the equation . The diagrams below will help to facilitate the process.
Using the distance formula the distance from to and the distance from to must be calculated. Remember that these distances are equal.
If the above steps are reversed, it can be confirmed that a solution of is equidistant from the focus and the directrix. If the parabola will open upward and if , the parabola will open downward.
Inverse relations of these parabolas are ones that open right if and left if . The and variables of change places and the standard equation becomes . All of these parabolas can be translated vertically and/or horizontally from the vertex thus changing the coordinates of the vertex and the standard equation. If the vertex is located at the standard equation of will be and that of will be
An ellipse can be defined as the set of all points in a plane such that the distances from two fixed points (foci) in the plane have a constant sum of 1. The line that passes through the foci is called the focal axis. The point on this axis that is midway between the foci is the centre of the ellipse and the points where the ellipse intersects the focal axis are the vertices.