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

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SUMMARY

The discussion centers on the phase shift of the sunrise equation, specifically why it equals 10 days. The sinusoidal equation used to predict sunrise times is D=180cos(2*pi*(t-10)/365)+372, where D represents days and t represents time. The phase shift of 10 days accounts for the positioning of the earliest and latest sunrise times, which occur on June 21 and December 21, respectively. This adjustment aligns the equation with the calendar year, starting from January 1.

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  • Knowledge of basic trigonometry, particularly cosine functions
  • Familiarity with time conversion, specifically converting time into minutes
  • Basic knowledge of the calendar year and its structure
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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?
 
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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.
 

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