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.
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.
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.