This question is hard to understand In London, the shortest day in 2014 was 21 June 2014, (the 172nd day of the year) when sunrise was at 06:37 (6.62 hours after midnight) and the longest day will be 23 December 2014 when sunrise will be at 04:50 (4.83 hours after midnight).
Develop a sinusoidal function to model the time of sunrise and use it to find the approximate day numbers when the sun will rise at 6:00.
 A: Here are some general guidelines, see if you can work out the details.
To start you should define variables, including units.  In this question an appropriate way to do that would be

let $y$ be the time of sunrise, measured in hours after midnight, on the $t$th day of the year.

So, you want a formula for $y$ in terms of $t$, involving sinusoidal functions, which reflects (at least approximately) the given data.  It would probably be a good idea to draw a sine function so that you have the general shape before your eyes.
You could begin with a function
$$y=\sin(at+b)\ .$$
Presumably the idea is that the data should repeat every year.  That is, the function should have period $365$.  This will give you the value of $a$.  Then, you need to use $b$ to "shift" the curve so that the maximum point is at $t=172$.
Now the vertical distance between the maximum and minimum values of your function is $2$.  But that's not what you want.  So, multiply by a suitable constant to get a function
$$y=c\sin(at+b)$$
which has the amplitude you want.
Finally, although the "vertical range" of this function is now of the right size, it is not in the right place (it will go from $-c/2$ to $c/2$).  So, find the constant you need to add to give
$$y=d+c\sin(at+b)$$
which has the maximum and minimum values you need.
Good luck!
