The sinusoidal equation will be Y=4.25 sin[(π/26)(t-11)]+12.25
How to compute the equation?
The following can be deduced:
Function Y=Asin(wt-Φ)+β
Amplitude A=(16.5-8)/2=4.25 hours
Vertical shift β= (16.5+8)/2=12.25
Period is 52 weeks, thus 52=2π/ω→ω=2π/52=π/26
The phase shift Φ can be gotten if we divide the whole year in 4 sub-intervals each interval would be :
[0,13], [13,26],[26,39],[39,52]
The summer solstice occurs at the end of week 24.
The winter solstice occurs at the end of week 51.
The maximum of sine function happens at the end of the first sub-interval [0,13]
Here, Φ is 11, thus, the function would be
Y=4.25 sin[(π/26)(t-11)]+12.25.
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