The independent variable is the index of the day of the year.
The dependent variable is the daylight time.
The amplitude of the function equals 2 hours and 53 minutes.
The period of the function equals 365 days.
The trigonometric function that describes the hours of sunlight for each day of the year is [tex]t(n) = t_{m} + (t_{h}-t_{l})\cdot \sin \left(2\pi\cdot\frac{n-1}{T} \right)[/tex], [tex]\forall \,n\in \mathbb{N}, n \ge 1[/tex].
The daylight time is a periodic function of the index of the day of the year, a year is equivalent to 365 days. In other words, the index of the day of the year represents the independent variable and the daylight time is the dependent variable.
Sinusoidal functions are the periodic functions that describe best the behavior of the daylight time throughout the year, which is represented by the following model:
[tex]t(n) = t_{m} + (t_{h}-t_{l})\cdot \sin \left(2\pi\cdot\frac{n-1}{T} \right)[/tex], [tex]\forall \,n\in \mathbb{N}, n \ge 1[/tex] (1)
Where:
- [tex]t_{m}[/tex] - Daylight time for equinox, in hours.
- [tex]t_{l}[/tex] - Daylight time for winter solstice, in hours.
- [tex]t_{h}[/tex] - Daylight time for summer solstice, in hours.
- [tex]n[/tex] - Index of the day of the year, no unit.
- [tex]T[/tex] - Number of days in a year (period), no unit.
- [tex]t[/tex] - Current daylight time, in hours.
Please notice that the amplitude of the function ([tex]\Delta t[/tex]) equals [tex]t_{h}-t_{l}[/tex].
If we know that [tex]t_{m} = 12.133\,h[/tex], [tex]t_{h} = 13.575\,h[/tex], [tex]t_{l} = 10.691\,h[/tex] and [tex]T = 365[/tex], then we derive the formula for New York City:
[tex]t(n) = 12.133 + 2.884\cdot \sin \left(2\pi\cdot \frac{n}{365} \right)[/tex] (2)
By direct inspection, we notice that the amplitude and the period of the function are 2 hours 53 minutes and 365 days, respectively.
We kindly invite to check this question of trigonometric functions: https://brainly.com/question/6904750
