Expert: Scotto - 11/21/2010

Question
In Prince Albert, Saskatchewan, the sun rises at 09:17 on December21 and at 04:35 on June 22. Because there is no daylight savings time in Saskatchewan, the time the sun rises or any other day can be predicted from a sinusoidal graph with a period of 365 days. How do I write a sinusoidal equation that relates the time the sun rises to the day of the year.
So,Dear Mr.Scotto,
         I jut want to make sure whether I am taking steps to solve this sum one after another properly or not.
So,can you please check my first step and sort out how can I make it better.After you have checked my first step,I would move down to the other:
 The general form of the sinusoidal is:
f ( x ) = a sin ( bx – c ) + d, for b > 0.
Hence, In order to find this sum's equation,we need to find its amplitude first.So,
   Amplitude would be the difference of two timings divided by 2 because they are the earliest and latest.Hence,
A = (9:17 - 4:35)/2 = 2:41.
Is this correct,Mr.Scotto? If it is alright,than I can move towards my next step.

Answer
Since there are 365 day in a year (365.25, to be technical about it).
The period of a sin wave is 2pi.
Multiply the number of days by 2pi/365, and this is b.
This gives the data the right period.

On a sin() wave, the max is 1 and the minimum is -1.
For what we have, 9:17 is the max and 4:35 is the min.
Converting both of these to minutes gives 557 and 275, respectively.
The difference between 557 and 275 is 557 - 275 = 282.
Since the sin() wave has a difference of 2, multiply by 282/2 = 141.
This is a.

Since the average of a sin() wave is 0 and the average time is
(557 + 275)/2 = 829/2 = 414.5, d is 414.5.

The last is c.  This is found by averaging the two days.
Note that December 21 is day number 355 in the year and
June 22 is, using the first five months and 22, 31+28+31+30+31+22,
which is around 173, so average 173 and 355 gives 264.
This also has to be adjusted by 2*pi/365, and then this is c.