Expert: Paul Klarreich - 7/18/2007

Question
Hey, I absolutely love solving math problems,this question however has given me a little trouble.  Please help.

   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.
Thanks in advance,
Triggy


Answer
Questioner:   Triggy
Category:  Advanced Math
Question:  Hey, I absolutely love solving math problems,this question however has given me a little trouble.  Please help.

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.
Thanks in advance,
Triggy
............................
Hi, Triggy,

Are you the famous British model/actress?  (Oh, wait -- that was Twiggy.  Never mind.)

The average of those times:  9:17 and 4:35  is 6:56.  That's your mean sunrise (which will occur on the equinoxes.)

So the sunrise will be a sine curve with a vertical displacement of  6:56.  (You should convert that to actual minutes -- 360 + 56 = 416 -- but we'll worry about that later.)

We have something like

SR = 416 +  A sin(....)

The amplitude of the variation is PLUS 2:21 and MINUS 2:21, because  9:17 = 6:56 + 2:21  and  4:35 = 6:56 - 2:21.

Our sine curve is now:

SR = 6:56 +  2:21 sin(fD + phi), where:

f = the frequency coefficient,

D = the Day of the year and is the VARIABLE. D = 0 on January 1, and D = 365 on Dec 31.

phi = the phase angle or horizontal displacement.

What about the frequency coefficient?  In 365 days, fd should equal 2pi.  

So 365f = 2pi and so  f = 2pi/365

SR = 6:56 +  2:21 sin(2pi D/365 + phi)

(We're getting there, said the dentist as the tooth began to come out.)

Last thing: the phase angle.  We want our sinusoidal curve to look like a sine curve.  No, I'm not saying a tautology, I mean it should start at zero and then increase to its maximum.

That happens at the Vernal Equinox, usually on March 21, which is day 80 of the year.  Therefore,

2pi(80)/365 + phi = 0

phi = - 2pi(80)/365

We're done:

SR = 6:56 +  2:21 sin(2pi D/365 - 2pi(80)/365)

which we can simplify to:

SR = 6:56 +  2:21 sin(2pi(D - 80)/365)