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I'm using Google Correlate tool to do some research and they have an example graph that is a sine wave that represents winter, the line was called "winter wave". http://www.google.com/trends/correlate/search?e=id:PaHT-seSlg9&t=weekly

I was curious if anyone could help me understand how to calculate a sine-wave where the peaks/troughs line-up with the calendar solstice (winter and summer).

Pat Grady
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You want a sine wave with a period of $365$ days, and you want the maximum to occur on day $173$ (June 22nd is the 173rd day of the year).

The formula for a sine wave with modified period and phase is

$$ y(t) = \sin(B(x - C)) $$

to change the period, set $B = 2\pi/365$. Then, recall that "regular" sine has its maximum at $\theta = \pi/2$, so we need to choose $C$ so that

$$ \frac{2\pi}{365}(173 - C) = \frac{\pi}{2} \\ C = 81.75 $$

Note that if we plug in $x = 81.75$, we will get the "middle" of the sine function (i.e. the place where it crosses the y-axis), and the the 81st day of the year is March 21, roughly corresponding to the spring equinox.

Further, you should check that when you plug in the winter solstice (first figure out what # day of the year it is), you should get the minimum of the sine.

BaronVT
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