Math and Sound Teacher Notes

Grade 12

Trigonometric Identities

Paul, Stanley, Erwin, and Gigi

Purpose: To show the trigonometric identity:
sin(2πf1t)+sin(2πf2t)=2 cos(2π (f1-f2)t/2)sin(2π (f1-f2)t/2)
in an auditory manner.

Students will hypothesis, predict and test which part of the wave is associated with the pitch and which part is associated with the beat or the volume.

Description of Activities: Students will be using Audacity (a free program) to generate sinusoidal sound waves. Playing two waves together is like playing the sum of the two waves. (This shows the additive property of waves.) Students will be investigating the beat phenomenon and the trigonometric identity: sin(2πf1t)+sin(2πf2t)=2 cos(2π (f1-f2)t/2)sin(2π (f1-f2)t/2) . They will be following their worksheet to guide them through the activity.

Sources: http://en.wikipedia.org/wiki/Beat_(acoustics)

Estimated time: 1-1.5 Classes or 1 Class and the rest is due for homework

Students are required to produce a page of answers and supporting graphs. Students will conclude cos(2π (f1-f2)t/2) is the part responsible for the amplitude (and thereby the beating) of the wave. The pitch of the wave is the part sin(2π (f1-f2)t/2) .

Marking Criteria: Students will be marked on the correctness of their answers. Students will also be given a mark on their participation in class out of a 4 point scale worth 25% of their total mark.

1- Did not participate

2- Participated 20% of the time

3- Barely meets expectation for participation

4- Meets expectation for participation

5- Exceeds expectation for participation