October 05, 2006

Scribe Post: Trigonometric Modeling

Hello everyone! This is Richard and I'm your scribe for today!

After the quiz on transformation, a new lesson was discussed to us. It was the Trigonometric Modeling which has something to do with circular functions and transformations. Mr. K gave us this example:

For a Saskatchewan town, the latest sunrise is on December 21 at 9:15 am. The earliest sunrise is on June 21 at 3:15 am. Sunrise times on other dates can be predicted using a sinusoidal equation.

(Note: There is no Dalight Saving Time in Saskatchewan)

(a) Sketch the graph of the sinusoid described above.

(b) Write two equations for the function; a sine & cosine equation.

(c) Use your equation to predict the time of sunrise on April 6.
(d) What is the average sunrise throughout the year?
(e) On what day will the sun rise at 7:00 am?

To be able to answer these questions, you must identify the given.

Latest sunrise - Dec 21 (9:15 am)
Earliest sunrise - Jun 21 (3:15 am)

Convert times into hours and dates into dates of the year.

9:15 = 9 15/60
= 9.25

3:15 = 3 15/60

= 3.25

Jun 21

J = 31
F = 28
M= 31
A = 30
M= 31
J = 21
172 days

Dec 21

J = 31
F = 28
M = 31
A = 30
M = 31
J = 30
J = 31
A = 31
S = 30
O = 31
N = 30
D = 21

355 days

Now let the hours be your y-coordinates and the days of the year as the x-coordinates. Let 9.25 hours be the max and 3.25 hours as the min. To determine the sinusoidal axis, get the average of the max and min.
9.25 + 3.25
__________ = 6.25 hours

You can now start to plot the graph.

( Note: This graph is not accurate)

Since the graph has only a half wave, It means that it only has a half period. A full period makes one wave. By looking at the garph, you can easily get the period.


Half Period = 355 days - 172 days

= 183 days

Full Period = 2 (183 days)

= 366 days

Quarter Period = 1/2 (183 days)

= 91.5 days

As soon as you know the period, you can now complete the graph. You have to draw the other half of the wave. To obtain the x-coordinates where the other max lies subtract 172 to the half period. So -11 is the location of the other max.

By also knowing the period, you can determine the day where the average sunrise time occurs.
You already know that the max and min has the same measure from sinusoidal axis. Since the point from the sinusoidal axis to the max or min is equal to quarter period, you can get the value of that point by getting the average of its max and min.

First point

max - 11
min 172

-11 + 172
________ = 80.5 days

Second point

max 355
min 172

355 + 172
________ = 263.5 days

So the average sunrise will fall on the 80th 1/2 day and 263 1/2 day.

At this point, we had figured out the answers in A and D. Until, kriiiiiinnnnggg! The bell rang. Our class period is over. We called it a day. JhoAnn will now continue the rest of the answers. I hope you gain something to my scribe.

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