If you play a guitar this might be interesting... [b]Key Terms[/b]: Trigonometric, Frequency, Coordinate Plane. [b]Purpose[/b]: To explain how sound waves determine the shape of chords on a guitar. [b]Introduction[/b]Sound waves are represented by an oscillating curve. They can be approximated by trigonometric functions. The sound produced by each note on a guitar produces a frequency that can be graphed in a coordinate plane.
[b]Activity[/b] The period of a sine curve is the length across the axis it takes to complete a full cycle. [list=8] [*]Analyze the period of each equation. [*]Write the period for each equation. [*]Analyze the graph of each equation again. [*]Looking at the red equation with the longest period, we can see that it completes a full cycle at x=6 and x=12. It also completes a full cycle at x=18,24,30... This is because the period of the red equation is 6. [*]The period of the blue equation is 4. Write a few multiples of the period. [*]The period of the green equations is 3. Write a few multiples of the period. [*]Compare the multiples of each equation. Write the lowest common multiple of all three equations. [*]What do you notice about the graph at this point on the x-axis? [/list] [b]Reflection[/b] A guitar chord is formed by playing the 1st, 3rd, and 5th note in a scale. Let's say that the red equation represents the 1st note, the blue represents the 3nd note, and the green represents the 5th note. Make a conjecture about how this parallels what happens on a guitar.