Daylight function for 40 °N Verify that the function $D(t)=2.8\(\sin\)(\(\frac{2\pi}{365}\)(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset. It has a period of 365 days.

Verified step by step guidance

1. Identify the general form of a sinusoidal function, which is typically given by A \(\sin\)(B(t - C)) + D, where A is the amplitude, B affects the period, C is the phase shift, and D is the vertical shift.
2. In the given function D(t) = 2.8 \(\sin\)\(\left\)(\(\frac{2\pi}{365}\)(t-81)\(\right\)) + 12, compare it to the general form to identify the values of A, B, C, and D. Here, A = 2.8, B = \(\frac{2\pi}{365}\), C = 81, and D = 12.
3. The period of a sinusoidal function is determined by the coefficient B in front of t. The formula for the period is \(\frac{2\pi}{B}\).
4. Substitute B = \(\frac{2\pi}{365}\) into the period formula: \(\text{Period}\) = \(\frac{2\pi}{\frac{2\pi}{365}\)}.
5. Simplify the expression for the period to verify that it equals 365 days, confirming that the function has the desired period.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and describe relationships between angles and sides of triangles. In the context of periodic phenomena, like daylight hours, the sine function models the cyclical nature of these changes over time, allowing us to predict values based on the angle of the input.

Periodicity

Periodicity refers to the repeating nature of a function over a specific interval. In this case, the function D(t) has a period of 365 days, meaning it repeats its values every year. Understanding periodicity is crucial for analyzing functions that model seasonal or cyclical behaviors, such as daylight duration throughout the year.

Phase Shift

Phase shift is a horizontal shift in the graph of a periodic function, affecting where the cycle begins. In the function D(t), the term (t - 81) indicates a phase shift, which adjusts the starting point of the sine wave. This is important for accurately modeling real-world phenomena, such as when daylight hours begin to increase or decrease throughout the year.