Modeling with Sine and Cosine Curves

Mth 111 Mathematics as a Human Pursuit Lab

Put your name here:

Follow instructions -- including typing in answers to questions. Remove output (Choose Edit-Remove Output), save (File-Save), email as an attachment when completed. You should print this or save to view on computer to prepare for your next test.

Execute the following two commands (Place cursor anywhere on the command and press Enter.)

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnRitGOkY9

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

Read Step 1 in the Analysis of your Sound Experiment. Here the experiment tells us that Logger Pro will fit a general sine curve to our sound wave data, where the form of the curve is:

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

Here A is the _________________ of the sound wave. B can be used to determine the frequency f with the formula B = 2*Pi*f where f is the frequency (or f = 1/period)

Thus when we know B, we can calculate the frequency as

____________.

Note: We will use E instead of D since D is protected symbol in Maple.

Note that \317\200 is an irrational number and is equal to the area of a circle with radius one. To see its value to 100 decimal places, execute the following command (Maple is case sensitive so the P in Pi must be capitalized):

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

1. Plot the standard sine curve y = sin(x) by executing the Maple command below.

This curve has a period = __________, amplitude = ______, and frequency = _________.

We plot the sine curve below for the range 0 to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtSSNtbkdGJDYkUSI0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy1GLDYlUScmIzk2MDtGJy8lJ2l0YWxpY0dGQEY3RjdGK0Y3RitGNw==:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUSNwMUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0Yr

2. Replace the WHAT below the appropriate value for A define a sine curve with amplitude 2:

3. Display the this curve with the standard sine curve:

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

Below is a plot of the function 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

What is the amplitude, period and frequency of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnRjkvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZQLUkobWZlbmNlZEdGJDYkLUYjNiZGKy1GIzYmLUkjbW5HRiQ2JFEiMkYnRjktRj02LVExJkludmlzaWJsZVRpbWVzO0YnRjlGQEZCRkRGRkZIRkpGTEZORlEtRiw2JVEieEYnL0Y3USV0cnVlRicvRjpRJ2l0YWxpY0YnRjlGK0Y5RjlGOUYrRjlGK0Y5?

Amplitude = _____ Period = ______ Frequency = ______

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVElcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYsLUYsNiVRJHNpbkYnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy1GNjYkLUYjNiUtSSNtbkdGJDYkUSIyRidGPy1JI21vR0YkNi1RIn5GJ0Y/LyUmZmVuY2VHRj4vJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGZ24tRiw2JVEieEYnRi9GMkY/LUZKNi1RIixGJ0Y/Rk0vRlBGMUZRRlNGVUZXRllGZW4vRmluUSwwLjMzMzMzMzNlbUYnRmpuLUZKNi1RIj1GJ0Y/Rk1GT0ZRRlNGVUZXRlkvRmZuUSwwLjI3Nzc3NzhlbUYnL0ZpbkZnby1GRjYkUSIwRidGPy1GSjYtUSMuLkYnRj9GTUZPRlFGU0ZVRldGWS9GZm5RLDAuMjIyMjIyMmVtRidGaG4tRkY2JFEiNEYnRj8tRko2LVEnJnNkb3Q7RidGP0ZNRk9GUUZTRlVGV0ZZRmVuRmhuLUYsNiVRJSZwaTtGJ0Y9Rj9GPw==

4. Below we plot the cosine curve:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUSNwM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0Yr

5. Display both curves together ( LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEkc2luRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnRjkvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZQLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEieEYnL0Y3USV0cnVlRicvRjpRJ2l0YWxpY0YnRjlGOUY5RitGOUYrRjk= and cos(x) ):

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

6. Below we plot the curve y = sin(x+C) with C = \317\200/2. How does 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 differ from the standard sine curve?

Answer: _________________________

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUSNwNEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0Yr

7. Below we plot the sine curve and the cosine curve with a horizontal shift to the right by _________.

We conclude that sin(x) = cos(x - __________).

LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmNyQtSSRjb3NHRiQ2IywmSSJ4R0YnIiIiSSNQaUdGJSMhIiIiIiMtSSRzaW5HRiQ2I0YuL0YuOyIiISwkRjAiIiUvSSZjb2xvckdGJzckSSdzaWVubmFHRidJJHJlZEdGJy9JKmxpbmVzdHlsZUdGJzckSSZzb2xpZEdGJ0klZGFzaEdGJw==

8. Here we display a standard sine curve along with a sine curve with parameters A, B, C, E -- Try different values for A, B, C and E in the command below to see how the graph changes. Explain what you see.

Answer:

Which (A,B,C, or E) represents the

Amplitude _________

Horizontal Shift _________

Vertical Shift _________

Angular Frequency -- Calculated as 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSIxRidGQEYrLUY9Ni1RIjtGJ0ZARkIvRkZGOEZHRklGS0ZNRk8vRlJRJjAuMGVtRidGVA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSNtbkdGJDYkUSIxRidGQEYrLUY9Ni1RIjtGJ0ZARkIvRkZGOEZHRklGS0ZNRk8vRlJRJjAuMGVtRidGVA==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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJFRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIxRidGPi1GOzYtUSI7RidGPkZAL0ZERjZGRUZHRklGS0ZNL0ZQUSYwLjBlbUYnRlI=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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

9. The pink graph below represents a sine curve -- Answer the questions below for this curve

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

10. For the pink sine curve above:

a) What is its amplitude?

b) What is its period?

c) What is its frequency?

d) What is its angular frequency?

e) What is its horizontal shift?

f) What is its vertical shift?

12. Enter below the correct values for A, B, C, E to model the curve above -- compare your plot to make sure it is correct.

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

13. For the equivalent cosine curve you only have to change one of the above parameters:

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

14. Beats: When two sound waves combine the resulting sound wave curve is the alegbraic sum of the two individual curves. Consider two sound waves represented by the functions:

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 ( plotted in red ) and 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 ( plotted in green )

and their algebraic sum

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(plotted in blue).

What is the frequency of the red curve? ___________

What is the frequency of the green curve? ___________

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

Plotting the algebraic sum by itself, we can definitely see the "beats". This curve is periodic with period = _________ and frequency = ______. Its maximum amplitude is ______.

Check to see that your results support the mathematical fact that the frequency of the beat curve (blue) should be the difference in the frequency of the two individual curves (red frequency - green frequency).

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