Sinusoids: Applications and Modeling

  • Objective
  • Level
  • Prerequisites
  • Platform
  • Instructor's Notes
  • Credits
  •  
    Objective The objective is to actively involve students via the software simulations so that the determination of the sinusoidal model has a geometric flavor that complements the algebraic tools stressed in texts. This approach also introduces a modeling aspect since in some situations we may only be able to obtain a "close" approximation to the actual curve or data.

    Level: This demo is appropriate for precalculus,  calculus, or introduction modeling classes.

    Prerequisites: Students should be familiar with basic trigonometric functions and the features of graphs of sinusoids y = a*sin(b*x) + c and y = a*cos(b*x) + c, including period, amplitude and vertical shifts. (For a review of this material see texts on precalculus or calculus; in addition, there are websites, see the list of auxiliary resources below, with reviews. For a collection of animations showing features of trigonometric functions and sinusoids click here.)

    Platform: This demo includes software that illustrates applications and modeling opportunities using Excel and Java applets. All routines can be downloaded or executed from within this demo.

    Instructor's Notes:

    General Comments: Many textbooks include exercises in which students are asked to determine the coefficients a, b, and c of a sinusoid of the form y = a*sin(b*x) + c or y = a*cos(b*x) + c. Such exercises are static, requiring the use of algebra, and often hold little appeal for students. This demo presents some applications like those in texts and others which involve real-world data. The objective is to actively involve students via the software simulations so that the determination of the sinusoidal model has a geometric flavor that complements the algebraic tools stressed in texts. This approach also introduces a modeling aspect because in some situations we are approximating real world phenomena with mathematical functions.

    Basically sinusoids are graphs of wave forms, hence any phenomena having periodic behavior or wave characteristics can be represented by trigonometric functions or (approximately) modeled by sinusoids. This includes many simple actions such as a pendulum, a child's swing, motion of an engine's piston-crankshaft, a Ferris wheel, tides, blood pressure in the heart, hours of daylight through out a year, (visible) shape of the moon, seasons, and sounds.

    A simple example of sinusoids are a person's biorhythms. Proponents of biorhythms claim our daily lives are significantly affected by rhythmic cycles and that these cycles can interact to indicate active and passive phases in the physical, emotional and mental aspects of humans. If you do a web search to find biorhythm software, it will produce a set of waves as illustrated in Figure 1. People who use biorhythms don't claim that biorhythms predict or explain events, only that biorhythms suggest how we may deal with them.

    Figure 1.

    Another very familiar example of an action which can be modeled by a sinusoid is the motion of a pendulum. See Figure 2. Plotting time versus the angle the pendulum arm makes with a vertical line that indicates the position of the pendulum at rest generates a sinusoid.

    Figure 2.

    A java applet for a pendulum is available at http://www.phy.ntnu.edu.tw/java/Pendulum/Pendulum.html and a Geometer's Sketchpad file by clicking here. The Sketchpad file asks that students make timing measurements using a stopwatch to use in determining the period.

    Modeling Sinusoids Graphically: After a discussion of the role of the coefficients of sinusoids y = a*sin(b*x) + c and y = a*cos(b*x) + c from an algebraic perspective it is helpful to reinforce this with geometry. For a collection of animations showing features of trigonometric functions and sinusoids click here. Figures 3 and 4 show typical frames of animations in this collection. Click on the figure number to see the accompanying animation.

    Such animations linked to a written assignment have helped students connect the standard algebra approach with the geometric significance of the coefficients. (Click here for an example.) A further step is to provide a tool for students to model a given sinusoid by determining the coefficients interactively using sliders. Two such tools (in a Java applet and Excel spreadsheet) are available in this collection that contain 20 problems. A typical problem in the Java applet is shown below. Move the sliders to develop a model of the form y = a*sin(b*x) + c.

    <<This is an interactive example.>>

    Click here to execute the general applet with its selection of problems.

    If you have Excel you can click here for the sine problems and here for the cosine problems to execute or download a file that has the same functionality as the applet. The graphics in the Excel file are better suited for projection in a classroom.

    An Engine Sinusoid: The periodic rotation of the piston-crankshaft assembly in an engine generates a sinusoid when we plot the angle of rotation of the crankshaft vs. the distance from the piston to the center of the circle as illustrated in the animation in Figure 5.

    Figure 5.

     

    If the radius of the circle is changed, then the sinusoid also changes. A Geometer's Sketchpad file submitted by David Roth suggested this connection to sinusoids and can be downloaded by clicking here. We also have an Excel file that simulates the engine and the resulting sinusoid. As a slider is dragged to simulate the rotation about the circle a sinusoid is generated. The user can change the radius of the circle. There is an option to then model the resulting sinusoid using the same technique discussed in the previous section on Modeling Sinusoids Graphically. By assigning students or teams different values for the radius this application could be used as an assignment for outside of class. To execute or download this Excel file click here. Click on the thumbnail below to see the style of the Excel screen.

     

    Ferris Wheel: At the Columbian Exposition in Chicago in 1893 George Ferris, a bridge builder from Pittsburgh, Pennsylvania, displayed the engineering highlight of the Exposition, the Ferris Wheel. His marvelous achievement has endured for more than a century and entertained countless people over the years. For more historical information see either of the following sites:
    http://www.hydeparkhistory.org/newsletter.html
    http://columbus.gl.iit.edu/dreamcity/00024024.html

    A Ferris wheel appears in a common textbook problem involving sinusoids where students are asked to determine the height of a rider above the ground. To involve students without actually going to a Ferris wheel at an amusement park, we have a simulation in Excel that lets them rotate  the wheel and change the radius of the wheel. (Ferris's original wheel had a radius of 125 feet.) As they simulate the rotation of the wheel a sinusoid is generated as a plot of the angle of rotation from the vertical vs. the height of a (particular) car above the ground. We illustrate this with the animation in Figure 6 where the height of the red car is of interest.

    Figure 6.

    We then present an option to model the resulting sinusoid using the same techniques discussed in the section on Modeling Sinusoids Graphically. By assigning students or teams different values for the radius this application could be used as an assignment for outside of class. To execute or download this Excel file click here. Click on the thumbnail below to see the style of the Excel screen.

    Engine & Ferris Connection: A good writing assignment is to have students or teams of students compare the "ideas" involved in generating the sinusoids in the engine demo and the Ferris wheel demo.

    Approximations to Data: The emphasis in mathematics on real-world applications often means that mathematical modeling must be used. In the previous activities in this collection we could often get "the" correct answer or very near it, depending on how fine an increment was used by the sliders that controlled the values of the coefficients. In the demos that follow this will not generally be the case. Given the simple type of sinusoid that we employ in the model we will need to be satisfied with an approximation that "looks pretty good" for the data presented.

    Hours of Daylight: The number of hours of daylight throughout the year is related to the periodic change in seasons. The corresponding changes in the tilt of the earth also causes the number of daylight hours to change. Because such changes are periodic from year to year the number of daylight hours on a given day is (nearly) the same. Hence a graph of the per day change in daylight hours is some sinusoid. This sinusoid is not the same for all locations on the earth. The shape of the sinusoid varies by latitude. We can see this from the following table.

    (This table is part of a very nice site was available from
    http://encarta.msn.com/media_701500905_761563211_-1_1/Hours_of_Daylight_by_Latitude.html when this demo was constructed; the link is no longer active.)

    Notice that at the equator the number of hours of daylight on the 15th of each month appears constant while it is radically different at the poles. In between the equator and poles the shape of the sinusoidal changes for daylight hours will more nearly be like those you have seen in other applications. To develop a simple sinusoidal model for the hours of daylight for a given city we need to know its latitude. We have an Excel program that contains latitude data for 60 cities worldwide for which we can develop a (simple) sinusoidal approximation to the hours of daylight curve for that city. For example, Nashville, Tennessee is at 36.1 North latitude and its hours of daylight curve for a year appears as shown in Figure 7.

    Figure 7.

    To build a model this curve we first rescale the data so that  the period is 2*pi and then we use sliders to control the coefficients in the sinusoid f(x) = a*cos(b*x) + c. The blue curve in Figure 8 is from the expression -2.2*cos(1.05*x) + 12.27 which provides a very good approximation.

    Figure 8.

    To execute or download this Excel file click here. Click on the thumbnail below to see the style of the Excel screen.

    In using the hours-of-daylight demo it is instructive to have students or teams of students develop models for pairs of cities. One near the equator and another quite a ways away from the equator. Cities near the equator can be modeled quite well as can those in middle latitudes. However, as you near the poles the simple sinusoidal model is not robust enough as we have implemented it. For example, the algorithm we adopted to generate the data when given information for Tronso, Norway, which is about 69 degrees North latitude, produces the hours of daylight curve in Figure 9. Obviously the technique used to generate this curve is flawed as we get near the poles.

    Figure 9.

    When making assignments, carefully choose pairs of cities. Another bit of caution. More sophisticated modeling procedures can be used, including least squares and Fourier approximations. Also aliasing for trigonometric models must be dealt with. Such topics are beyond the scope of this demo collection.

    Energy Bills: A very nice example of sinusoids "at home" comes in the form of monthly energy bills. In the article by Cathy Schloemer, "I Found Sinusoids in My Gas Bill", Mathematics Teacher, Vol. 93, No. 1, January 2000, pp. 10-12 she shows how to develop a model using an "eye-balled" approach as well as providing an explanation of how to use the TI-83 calculator regression procedure to approximate sinusoidal data based on month vs. average monthly temperature and another based on month vs. gas used. The modeling of such discrete data is very appropriate for a wide variety of situations.

    We provide Excel routines which let users interactively develop models for the data given in this article. In these routines we set up the function f(x) = a*sin(b*(x - k)) + c where the values of a, k, and c can be chosen by moving sliders and b is fixed at pi/6 since we imagine the period of the data to be 12 (months). Two curves are shown on the same graph: one for the original data set with the data points indicated by red markers, and a second curve corresponding to f(x) = a*sin(b*(x - k)) + c for the current setting of the sliders with the values along this curve at the months indicated by blue markers. See Figure 10.

     

    Figure 10.

    To aid in determining a good approximation the routines display a table with original and current model values for each month and a graph of a square whose area is equal to the sum of the squares of the differences between these monthly values. Thus we easily can see a measure of goodness for the current model. The smaller the area of the square displayed the better the approximate curve in what is termed the least-squares sense. However, since the sliders are incremented using a fixed step size we will most likely not construct "the" least squares model, but we can come close.

    For both data sets given in the paper the Excel routines can, after some experimentation, obtain quite good models. These models vary a bit from those developed in the article as expected. To run or download these Excel routines, click here for the month vs. temperature model and here for the month vs. gas usage model. Click on the thumbnail below to see the style of the screen design for these routines.

    A possible class project is to replace the data from the article with that from the students' homes. This requires changing only 12 values in the spreadsheet.

    Optional Advanced Comments: The TI-83 regression results given in the paper generate a solution using optimization techniques that obtain a least squares model varying all four coefficients in the function f(x) = a*sin(b*(x - k)) + c. As suggested by optimization practitioners it is recommended that the results of several non-linear optimization routines be compared. We found that the sum of the squares of the differences between the original monthly values and values obtained from the TI-83 regression for the optimization of the month vs. gas usage data was approximately 164.96. (For the equation from the TI-83 see Figure 8 in the paper.) For comparison we used Matlab's lsqcurvefit function and found the model to be

    f(x) = 10.75904219464498 * sin(0.43152271130626 *x +  1.20836380096032 )+ 12.66569717166289

    with the sum of the squares of the differences between the original monthly values and values obtain from this model to be approximately 57.17. In fact, using the Excel routine described above for this data set we were able to "do better" than the TI-83 regression model, but not as good as the Matlab result.

    The Excel routines mentioned above can be changed to include a slider to control the b coefficient. With this added feature we should be able to get a better fit to the original data, but recall that the sliders were only designed to increment at fixed step values.

    Other Resources:

    For some Biorhythm Fun! sponsored by National Institute of
    Environmental Health Sciences go to http://www.niehs.nih.gov/kids/biorhythm/jvbio.htm


    The works cited here are also useful.

    At http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/trig/trig1.html
    is another site with general background information and nice explanations.

    At http://www.visualprosthesis.com/im2sound.htm has information on sounds and artificial scenes that may form the basis for student projects.

    At http://www.mindspring.com/~scottr/zmusic/ you can combine sine waves to produce musical tones and the human voice.

    At www.ti-edu.com/pdf/rmb.pdf is a description of a respiration monitor belt which is used to measure human respiration rates. Included are directions for setting up a computer or CBL system to monitor respiration to gather and process data to build sinusoidal models for respiration rates.

    At http://www.arcticice.org/daylight.htm is a nice site describing seasonal changes and hours of daylight.

    Using the U.S. Naval Observatory site at http://aa.usno.navy.mil/data/docs/RS_OneDay.html you can obtain the times of sunrise, sunset, moonrise, moonset, transits of the Sun and Moon, and the beginning and end of civil twilight, along with information on the Moon's phase by specifying the date and location. In addition you get longitude and latitude information.

    An applet for daylight hours and additional information is available at http://www.jgiesen.de/daylight/ .

    At http://curious.astro.cornell.edu/question.php?number=405 is a site that discusses "Why doesn't the length of each day change much around the solstices?"

    Download files: The following zipped files can be downloaded to get the software discussed in this demo. Just click on  the platform name.

    Geometer's
    Sketch Pad
    Excel Java Applets

    Comments:
         1. We recommend that when using our Excel files that you "enable macros".
         2. When using the Java applets the file jcm1.0-config.jar must be in the same folder as the applet.

    Credits:  The idea for this demo started with the Geometer's Sketchpad 'engine' animation submitted by David Roth after a discussion at the Georgia Mathematics Conference in 2003. Portions of this demo benefited greatly from discussions with Mark Yates at the McCallie School. This demo collection was constructed by David R. Hill who also created the Java and Excel files. The Java applets were constructed using portions of Java Components for Mathematics which was developed under NSF grant number DUE-9950473.

    This demo collection is included in Demos with Positive Impact with the permission of the contributors.


    DRH 7/21/2004          last updated 9/15/2010 DRH

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