I have to find a function

$$f(x)=a\sin(bx+c)+d$$

where $f$ is, in hours, the duration of a daylight. I know in the summer solstice, this duration – which is the maximum – is 14 hours and in the winter solstice (which is the minimum), 9 hours.

The question requests to be $x=0$ the spring equinox and each season lasts 90 days. Therefore

$$f(90)=14,\qquad f(270)=9$$

Using this data, I could find the values of $a,b,c$ and $d$. However, the duration of a year should be 365 days.

Since for $x=90$, $f$ has its max value, then

$$\sin(90b+c)=1\Rightarrow90b+c=\frac{\pi}{2}$$

and using the same idea

$$\sin(270b+c)=-1\Rightarrow270b+c=\frac{3\pi}{2}$$

My question is, since the year has 365 days, how do I work with this arcs? I mean, should I use the following equations instead:

$$90b+c=\frac{\pi}{2}\div365$$

$$270b+c=\frac{3\pi}{2}\div365$$

## Best Answer

I don't understand the suggested division in your last two lines; I think a different model is called for. One adapted model is $$f(x)=a\sin\left(\frac{360}{365}bx+c\right)+d.$$