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 NCB Deposit  # 26


The Students of CS 390
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The Two Most Studied Curves in Trigonometry

The Sine and Cosine Functions

An oscilloscope traces the
sound wave of a recorder.
oscilloscope

Background . . . .

Sine equation



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Warning:
This animation is an applet that requires the Java 2 plug-in. If you do not have this free software, go to <http://www.java.com/ > and click on "Get It Now."  Follow the on-screen instructions to download and install for YOUR browser.  Please note that the Java 2 plug-in is not available for Mac OS 9 applications, e.g. Netscape 4.7.

 
Significance

Many different physical phenomena may be measured by their wave motion.  Perhaps the best known examples are those of radio waves, television waves or microwaves.    Indeed, radio stations are named for their types of wave transmission.  For an AM station, the  amplitude of the wave is varied to carry different sounds.  An FM station has its  frequency modified.  Thus, AM stands for  amplitude modulation and FM represents frequency modulation.

The difference may be illustrated using the sliders on the above graphic.


Observation:

Any book on trigonometry will have these two, much studied functions.  However, in the physical sciences where math is often applied, the sine function, or cosine function may be "squared."   The  x value often represents time and the  y value, i.e., the function value, is the periodic signal being studied. 

In the case that the sine or cosine function is "squared,"  the student may recognize this as being similar to the  "piecewise" function, or "step" function studied in earlier algebra courses.

The more advanced student will also recognize the function as an expanded Fourier series with a periodic cycle.

Step function graph

Step function equation

Students will recognize a familiar 
piecewise function from Pre-calculus 
can easily be applied to time and a time 
function.

Links and Printed References
Animated Unit Circle  <  http://curvebank.calstatela.edu/unit/unit.htm  >
Another animated Unit Circle  < http://curvebank.calstatela.edu/sine/sine.htm  >
Square Functions and other Fourier applications for the advanced student.  See  < http://www.falstad.com/mathphysics.html  >.
John Johnson, Jr.,  "COLUMN ONE: Gravity's Field of Dreams," Los Angeles Times, June 10, 2006, p. 1.
Donald A. McQuarrie,  MATHEMATICAL METHODS  for Scientists and Engineers, University Science Books,  Sausalito, CA, ISBN 1-891389-24-6, 2003.
Waves on a Cosmic Scale

Waves and wave equations have significance far beyond the mathematics of the trig classroom.  In fact, the study of gravitational waves is currently one of the largest projects in experimental physics in the United States.  The project is called  LIGO for  Laser Interferometer Gravitational Wave Observatory.  In order to collect good data, a minimum of two large facilities had to be built.  One is in the State of Washington, and the other in Louisiana.

                          LIGO Hanford Observatory, Washington          California Institute of Technology 
                          LIGO Livingston Observatory, Louisiana           Massachusetts Institute of Technology 

LIGO Animations of gravitational waves <http://www.ligo.caltech.edu/LIGO_web/aaas0203/ >.

To view the animations you must have RealPlayer or RealOne Player installed on your computer.  You can download a free copy at < http://www.real.com >.  The link to download is located on the upper right hand corner of the page labeled as "free RealOne Player".  Once you have successfully installed the player you can visit the LIGO website.  Click on the "animations" link and select the appropriate speed that applies to your internet connection.  Most campuses use a LAN connection.

For more information please see LIGO :  <  http://www.ligo.org  > 

Photo: LIGO Gravitational Wave Observatory
Photo courtesy of Caltech, Pasadena, CA.

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Matthew Nelson
Jonathan Liu
Mahan Hajianpour