Why Does the Phase Shift in the Sunrise Equation Equal 10 Days?

[davedave]

At Estevan, Saskatchewan, the latest sunrise time is at 9:12am on Dec 21. The earliest sunrise time is 3:12pm on June 21. Sunrise times on other dates can be predicted from a sinusoidal equation. There is no daylight saving time in Saskatchewan and the period is 365 days. Convert the time into minutes and the date into days. That is, June 21 is 172 days and Dec 21 is 355.

The equation is given by

D=180cos(2*pi*(t-10)/365)+372

where D=days and t=time

Does anyone know why the phase shift is equal to 10?

 

[HallsofIvy]

Strictly speaking, the phase shift is 20pi/365. But the "10" is there because t represents the day of the year starting at Jan.1 and the base information, latest and earliest sunrises, is at Dec. 21 and June 21, 10 days earlier than the beginning and middle of the calendar year.