Adding Trigonometric Functions
Lesson Plan

Grade level: 11th or 12th
Time: 75 minute block or two 50 minute periods (better)

Objectives:

• Students will be able to manually add two sine waves together.
• Students will be able to describe the conditions that cause a beat or pulsing pattern.
• Students will be able to attribute the characteristics of a summation graph to each of the individual terms.
• Students will be able to demonstrate the significance of adding sound waves together for nonstandard waveforms (i.e. triangle, square, etc.)

Prerequisite Knowledge:

• Students will have previously studied properties of sine waves, and should be able to graph them and describe the parameters.
• Students will know terms such as period, frequency, amplitude and phase and how they relate to a graph of a sine function.

Materials:

• Student Worksheets (these will open in a new window)
• Computers with Internet access (and speakers)
• 3 different colored pens or thin markers

Introduction:

• We know how to graph trigonometric functions. We know about their properties. Now what happens if we want to combine two trig functions?
• Examples of y = x and y = 2x, and y = sin x and y = 2 sin x, point out grouping like terms.
• Cannot always group like terms.
• What is the relevance? Music. Guitars. Radio. Etc., Etc.

Activity:

• Student worksheet 1 - manual addition of sine waves. Constructive and Destructive interference examples should be straightforward. Emphasize using the table to get some data points. The rest should be filled in by looking at the original graphs. Final example will be more difficult because the pattern is not regular. Be sure to encourage the table, and model following the two original graphs point by point. (I.e. two positives = a bigger positive point. One positive one negative = somewhere around the axis.)
• Student worksheet 2 - using the Java applet to find patterns adding sine waves. Be sure to watch how they set things up. Look for double sine wave pattern in the first activity. For the second activity, the coefficient a should be near the coefficient for the other term, but not equal. May need to review period, wavelength, frequency terminology.
• Student worksheet 3 - sounds and sine waves. Wave files might be slow to download. First applet does traveling sine waves, find the beat pattern. What would it sound like? Second applet simulates fourier series additions (big scary words) to find out what kind of sine waves add up to a triangle wave. May need to work through terminology for this one.

Conclusions:

• Where else might wave addition come in useful?
• AM vs FM radio as an example
• We don't make these functions up for our health!

Extensions:

• Have students explore and explain what happens when you add sin (x) and cos (x) in the same fashion as the problems on worksheet 1.
• Have students explore a square wave that is symmetrical over the y-axis in the same manner that the triangle wave on worksheet 3 is explored.

Assessment:

• Collect students manual plots and their answers for worksheet two. This should give a good idea as to their technical skills at adding the functions.
• Engage in discussion with the students about the sound waves. Ask groups about their predictions before they listen to the sounds. Review their answers to the triangle wave problem.
• For a more formal assessment, students should be able to manually add two trig functions together on a test or quiz. When asked, they should also be able to talk about using sines and cosines to form other functions, especially in relation to sound.