CK-12 Trigonometry
Page 26
During a baseball game, a ball is hit into right field. The angle from the ball to home to base is . The angle from the ball to to home is . The distance from home to base is . How far was the ball hit? How far is the baseman from the ball?
Solve using the diagram below.
A pool player is preparing to make his final shot of the game and the cue ball is from the ball. The ball is from pocket and from the pocket . If the cue ball is from the pocket and from the pocket , which shot has the smaller angle?
Solve using the diagram below.
The military is testing out a new infrared sensor that can detect movement up to thirty miles away. Will the sensor be able to detect the second target? If not, how far out of the range of the sensor is Target ?
Solve using the diagram below.
An environmentalist is sampling the water in a local lake and finds a strain of bacteria that lives on the surface of the lake. In a one square foot area, he found bacteria. There are three docks in a certain section of the lake. If Dock is from Dock , how many bacteria are living on the surface of the water between the three docks?
A forest ranger in Tower spots a fire away at a direction east of north. If Tower is directly east of Tower , how far is the fire from Tower ?
Two bulldozers are pushing a large footing for a building at the same time. One bulldozer exerts a force of in an easterly direction. The other bulldozer pushes with a force of in a southerly direction. What is the magnitude of the resultant force on the footing?
What is the direction of the resultant force?
A pilot leaves the airstrip and travels at a heading of . Then, he travels at heading of . How far from the airstrip has he traveled and at what heading?
Review Answers
The horizontal distance is
The distance across the canyon is
The distance between the two stoplights is
The ball is hit The second baseman is away from the ball.
The shot to pocket has the smaller angle.
No, the sensor will not be able to detect the second target. It is out of the sensor’s range.
bacteria.
The fire is from Tower .
The magnitude is . The direction is .
He has traveled from the airstrip at a heading of .
Supplemental Links
http://www.coastal.edu/mathcenter/HelpPages/Handouts/oblique.PDF
Chapter 6: Polar Equations and Complex Numbers.
Polar Coordinates
Learning Objectives
A student will be able to:
Distinguish between and understand the difference between a rectangular coordinate system and a polar coordinate system.
Plot points with polar coordinates on a polar plane.
Introduction
Have you ever wondered how a surveyor is able to obtain accurate measurements of land that is neither rectangular nor flat? A device called a theodolite is used. This device is able to measure horizontal and vertical angles to determine exact land locations and features. Let us suppose that you are surveying a piece of land on which to build your dream home.You notice two distinct landmarks that indicate the boundary of your property.You see an oak tree away and to the left and an apple tree away and to the right. What is the length of your property? We will determine this answer later in the lesson.
Figure 6.1
Figure 6.2
The graph paper that you have used for plotting points and sketching graphs has been rectangular grid paper. All points were plotted in a rectangular form by referring to a perpendicular and axis. In this section you will discover an alternative to graphing on rectangular grid paper – graphing on circular grid paper.
Look at the two options below:
You are all familiar with the rectangular grid paper shown above. However, the circular paper lends itself to new discoveries. The paper consists of a series of concentric circles-circles that share a common centre. The common center , is known as the pole or origin and the polar axis is the horizontal line that is drawn from the pole in a positive direction. The point that is plotted is described as a directed distance from the pole and by the angle that makes with the polar axis. The coordinates of are .
These coordinates are the result of assuming that the angle is rotated counterclockwise. If the angle were rotated clockwise then the coordinates of would be . These values for are called polar coordinates and are of the form where is the absolute value of the distance from the pole to and is the angle formed by the polar axis and the terminal arm .
Example 1:
Plot the point ) and the point
Solution:
To plot , move from the pole to the circle that has and then rotate clockwise from the polar axis and plot the point on the circle. Label it .
Solution:
To plot , move from the pole to the circle that has and then rotate counter clockwise from the polar axis and plot the point on the circle. Label it .
These points that you have plotted have r values that are greater than zero. How would you plot a polar point in which the value of r is less than zero? How could you plot these points if you did not have polar paper? If you were asked to plot the point or you would rotate the terminal arm counterclockwise or . (Remember that the angle can be expressed in either degrees or radians). To accommodate , extend the terminal arm in the opposite direction the number of units equal to . Label this point or whatever letter you choose. The point can be plotted, without polar paper, as a rotation about the pole as shown below.
The point is reflected across the pole to point.
There are multiple representations for the coordinates of a polar point . If the point has polar coordinates , then can also be represented by polar coordinates or if is measured in degrees or by or if is measured in radians. Remember that is any integer and represents the number of rotations around the pole. Unless there is a restriction placed upon , there will be an infinite number of polar coordinates for .
Example 2: Determine four pair of polar coordinates that represent the following point such that .
Solution:
Pair 1
Pair 2
Using and
Pair 3
Using and
Pair 4
Using and
These four pairs of polar coordinates all represent the same point P. You can apply the same procedure to determine polar coordinates of points that have measured in radians. This will be an exercise for you to do at the end of the lesson.
Example 3: Did you forget about building your dream home?
What is the length of your property on which you are going to build your dream home? Before you can calculate this distance, you should represent your property to indicate the landmarks. Here is a sketch of what you know.
represents the apple tree is negative – clockwise from zero degrees
represents the oak tree is positive- counter clockwise from zero degrees
represents the pole
represents .
represents .
represents
represents .
If you join to this will create a side of and its length can be determined by using the polar distance formula which is the polar version of the Law of Cosines.
The length of your property is approximately .
Lesson Summary
In this lesson, we have explored an alternative method of graphing. We have plotted points with polar coordinates by using a polar grid form. We have also noticed that this has connections to previously learned topics like rectangular graphing, the Law of Cosines, and angles in standard position. In subsequent lessons, we will explore additional relationships between rectangular and polar graphing and extend these relationships to involve the world of complex numbers.
Points to Consider
How is the polar coordinate system similar/different from the rectangular coordinate system?
How do you plot a point on a polar coordinate grid?
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p; How do you determine the coordinates of a point on a polar grid?
How do you calculate the distance between two points that have polar coordinates?
Review Questions
Graph each point:
For the given point , list four different pairs of polar coordinates that represent this point such that .
Given and , calculate the distance between the points.
Review Answers
Using and
Using and
Using and
Using and
First Pair
Second Pair
Third Pair
Fourth Pair
The distance between the two points is approximately units.
Vocabulary
Polar coordinate system
A method of recording the position of an object by using the distance from a fixed point and an angle consisting of a fixed ray from that point. Also called a polar plane.
Pole
In a polar coordinate system, it is the fixed point or origin.
Polar axis
In a polar coordinate system, it is the horizontal ray that begins at the pole and extends in a positive direction.
Polar coordinates
The coordinates of a point plotted on a polar plane .
Sinusoids of One Revolution (e.g., limaçons, cardioids)
Learning Objectives
A student will be able to:
Graph polar equations.
Graph and recognize limaçons and cardioids.
Determine the shape of a limaçon from the polar equation.
Introduction
An unidirectional microphone is sensitive to sounds from one direction with the most common of these being the cardioid microphone. This name comes from the fact that the sensitivity pattern is heart-shaped.
Figure 6.3
Where have you seen pictures that display sound as it travels from different directions? If you think about animated captions, they are frequently drawn around a megaphone or a microphone. A polar coordinate system can also be used to represent the patterns of these frequencies. The pole represents the microphone and is used to locate the source of the sound that travels around the pole. The amplitude of the sound is the value of r. Later in the lesson, we will look more closely at this relationship by sketching a graph to represent the polar pattern.
Just as in graphing on a rectangular grid, you can also graph polar equations on a polar grid. These equations may be simple or complex. To begin, you should try something simple like or where is a constant. The solution for is simply all ordered pairs such that and is any real number. The same is true for the solution of . The ordered pairs will be any real number for and will equal . Here are the graphs for each of these polar equations.
Example 1: On a polar plane, graph the equation
Solution:
Example 2: On a polar plane, graph the equation
Solution:
To begin graphing more complicated polar equations, we will make a table of values for or in this case . When the table has been completed, the graph will be drawn on a polar plane by using the coordinates .
Example 3: Create a table of values for such that and plot the ordered pairs. (Note: Students can be directed to use intervals of or allow them to create their own tables.)
Remember that the values of are the values.
This is a sinusoid curve of one revolution.
We will now repeat the process for .
Example 4: Create a table of values for such that and plot the ordered pairs. (Note: Students can be directed to use intervals of or allow them to create their own tables.)
Remember that the values of are the values.
This is also a sinusoid curve of one revolution.
Notice that both graphs are circles that pas through the pole and have a diameter of one unit. These graphs can be altered by adding a number to the function or by multiplying the function or by doing both. We will explore the results of these alterations by first creating a table of values and then by graphing the resulting coordinates
Example 5: Create a table of values for such that and plot the ordered pairs. Remember that the values of are the values.
This sinusoid curve is called a limaçon. It has or as its polar equation. Not all limaçons have this shape-an inner loop. Some may curve to a point, have a simple indentation known as a dimple or curve outward. The shape of the limaçon depends upon the ratio of where is a constant and b is the coefficient of the trigonometric function. In example 5, the ratio of which is . All limaçons that meet this criterion will have an inner loop.
Using the same format as was used in the examples above, the following limaçons were graphed. If you like, you may create the table of values for each of these functions.
i) such that
which is but
This is an example of a dimpled limaçon.
ii) such that
which is
This is an example of a convex limaçon.
Example 7: Create a table of values for such that and plot the ordered pairs. Remember that the values of are the values.
This type of curve is called a cardioid. It is a special type of limaçon that has or as its polar equation. The ratio of which is equal to .
Examples 3 and 4 were shown with a measured in degrees while examples 5 and 7 were shown with measured in radians. The results in the tables and the resulting graphs will be the same in both units.
Now that you are familiar with the limaçon and the cardioid, also called classical curves, it is time to examine the polar pattern of the cardioid microphone that was introduced at the onset of the lesson.. The polar pattern is modeled by the polarequation . The values of and are equal which means that the ratio . Therefore the limaçon will be a cardioid.
Create a table of values for such that and graph the results.
This pattern reveals that the microphone will pick up loud sounds behind it but softer sounds in front.
What does this pattern tell you about the cardioid microphone?
Lesson Summary
In this lesson we have explored graphing polar equations - both simple and complicated. We have also become familiar with the various functions that model the different sinusoids of one revolution. These ideas will be utilized in further lessons to extend your knowledge of limaçons and transformations of these curves.
Points to Consider
How do you graph a polar equation?
What type of graph results from graphing a polar equation?
Is it possible to name type of classical curve without graphing the function? Justify your response
Review Questions
Name the classical curve in each of the following diagrams and be specific in your response.
Another classical curve is called a rose and it is modeled by the function or where is any positive integer. Graph and . Is there a difference in the curves? Explain. What role does play in relation to the graphs?
for
Review Answers
a limaçon with an innerloop.
a cardioid
a dimpled limaçon
for
In the graph of , the rose has four petals on it but the graph of has only three petals. It appears, that if is an even positive integer, the rose will have an even number of petals and if is an odd positive integer, the rose will have an odd number of petals.
Applications, Trigonometric Tools
Polar Coordinates and Polar Equations
Learning Objectives
A student will be able to:
Understand real-world applications of polar coordinates and polar equations.
Introduction
In this lesson we will explore examples of real-world problems that use polar coordinates and polar equations as their solutions.
Example 1:
The pole or origin is the black dot at the center of the clock face. The polar axis, the hour hand, is three units in length and extends from the po
le to the number three. The minute hand is four units in length. What are four possible polar coordinates of the tips of the hour hand at 1:00 o’clock such that ?
Solution:
We have the polar coordinates for point T. Three other pairs of polar coordinates for T are:
Using and
Using and
Using and
Example 2: A local charity is sponsoring an outdoor concert to raise money for the children’s hospital. To accommodate as many patrons as possible, they are importing bleachers so that all the fans will be seated during the performance. The seats will be placed in an area such that and , where is measured in hundreds of feet. The stage will be placed at the origin (pole) and the performer will face the audience in the direction of the polar axis .