Answer to Question #103430 in Trigonometry for tamara

Question

Can hours of daylight data be modeled as a sinusoidal function for every location on earth? Explain.

Answer 1

As we know that earth is completing one round around the sun in $365\dfrac{1}{4}$ days on average.

The longest day is June 21, on this day sun rise at 4:47AM and sunset at 7:38PM, so the daylight hour is 14hour 51 min. So daylight hour is 14.85hour.

And on the shortest day is December 21, on this day sun rise at 7:24AM and sunset at 4:54PM, so the daylight hour is 9.48 daylight.

So the amplitude of the daylight hour = $\dfrac{14.85-9.48}{2}=2.685$ hours

mid line of the =$\dfrac{14.85+9.48}{2}=12.165hours$

The average time period of the cosine function =$365$

So, the cosine function without the horizontal shift $f(x)=2.685\sin(\dfrac{2\pi}{365}t)+12.175$

So, if we are taking any random day for example Jan 14,

It is the 207th day after the longest day.

So, $f(207)=2.685\sin(\dfrac{2\pi}{365}207)+12.175=9.734hours$

Which is reasonable as compared to 21Dec, from the shortest day, which is just 3 weeks near to the shortest day.