NB this is my way of approaching the problem, it may not necessarily be correct so dont take my answer to be perfect

ok we need a sine function the most general form of this possible is

first of all it would need a period of 365 days (assuming we havn't got a leap year) therefore as

looking it up on wikipedia the longest day of the year is on 21st June, using your numbering system this would be day 172

also the shortest day is on 21st December, so that would be day 355.

we know the function must be a maximum on day 172, so therefore

solving this for alpha gives us

ok were half way there!

the longest day of the year is 14 hours 42 mins = 14.7 hours

therefore when the function is a maximum y=14.7

sine has is max value of 1 so

the shortest day of the year is 9 hours 18 mins=9.3 hours

sine has a min value of -1 so

solving the two equations gives us A=2.7 and B=12

so the final equation is