Sounds like fun. Here we go.
We need to come up with A, k, and c for the equation y = Asin(kx+c).
Some basic information is that there are 365 days in a year. The least amount of daylight is on the first day of Winter - around December 21. The most is around June 21. It will be even amounts on the equinox dates - March 21 and September 21.
With this info, we can examine amplitude, period, and phase shift.
Amplitude - Let's assume we get three extra hours on the first day of summer and three fewer hours on the first day of winter. So A = 3. That was easy.
Period - Since Period = (2pi)/k and the daylight cycle is 365 days, so 365 = (2pi)/k. Therefore k = (2pi)/365 = 0.0172
Phase Shift - It would have been convenient if the spring equinox fell on January first. There would have been no phase shift. Instead, it falls about 80 days later. Phase shift = -c/k, So 80 = -c/0.0172. We get c = -1.376.
Our equation can be written y = 3sin(0.0172x-1.376). The x stands for the number day of the year, such as for January 12, x = 12. For February 1, x = 32. The y stands for the amount of extra or less sunlight. I tried it out for a few values and it seems to work. On December 21 we have three fewer hours. On January 1, we have 2.94 fewer hours - a slight improvement.
We could change the formula so it could stand for the full amount of daylight by tacking on a +12 to the end: y = 3sin(0.0172x-1.376)+12
