Answer to Question #94848 in Trigonometry for may

Question

In 2001, Windsor, Ontario will receive its maximum amount of sunlight, 15.28 hrs, on June 21, and its least amount of sunlight, 9.08 hrs, on December 21.

a) Due to the earth's revolution about the sun, the hours of daylight function is periodic. Determine an equation that can model the hours of daylight function for Windsor, Ontario.

b)On what day(s) can Windsor expect 13.5 hours of sunlight?

Answer 1

a)

It is known that f(x) is called a periodic function with period T if f(x) = f(x + T) for each x.

sin(x) is a periodic function with period $2\pi$.

We can simulate the required process in the form of harmonic oscillation with bias: F(t) = C + A*sin($\omega t + \phi)$.

The first step is to scale the sine function vertically.

The amplitude - the maximum deviation from equilibrium - of the classical sine function is 1.

Maximum value of the new function in 15.28, minimum value is 9.08.

Then the average value is (Max + Min) / 2 = 15.28 + 9.08 / 2 = 12.18. Actually, this is a vertical bias - C.

The new amplitude is A = Max - Average = 15.28 - 12.18 = 3.1.

The next step is to scale the sine horizontally. As it is said above, the period os the sine function is $2\pi$.

The period of the target function is 365. In order to move to a new period for the sine function it is necessary to multiply the argument by $2\pi / T_1$, where $T_1$ is a new period (in our case $T_1 = 365$).

Then now the function takes the following form: $F_1(t) = 12.18 + 3.1* sin(2\pi * t / 365)$

Actually we can stop now. But in addition it is possible to choose the starting point $t_0$ and add a horizontal bias.

For example, let suppose that you wish to define $t_0$ such that it corresponds to the January, 1.

By the condition of the problem, maximum value is reached on June 21 which is the 172-nd day of the year.

At the same time $min_{t>0} (argmax(F_1(t)) = 92.$ That's why the horizontal bias for argument t should be added:

t + 92 - 172

The final function form is $F(t) = 12.18 + 3.1 * sin(2\pi * (t + 92 - 172) / 365$

$F(t) = 12.18 + 3.1 * sin(2\pi *\frac{ (t - 80)} { 365})$

b)

If t equals 105 (April 15) F(t) approximately equals 13.52.

If t equals 236 (August 24) F(t) approximately equals 13.50.