Determine the amplitude and period of a periodic context
Model periodic behavior with sinusoidal functions
Write both a sine and cosine function to model the same periodic behavior
Temperature Variation in a Desert City
The temperature in Phoenix, Arizona varies throughout the day. On a typical summer day, the temperature reaches a maximum of 105°F at 4:00 PM and a minimum of 75°F at 4:00 AM. The pattern repeats every 24 hours.
Find a sinusoidal function of the form [latex]T(t) = A\cos(B(t - C)) + D[/latex] that models the temperature, where [latex]t[/latex] is hours after midnight.
What is the temperature at 10:00 AM?
Write an equivalent sine function to model the same temperature pattern.
Calculate [latex]B[/latex] from the 24-hour period: [latex]B = \frac{2\pi}{24} = \frac{\pi}{12}[/latex]
Find phase shift: Since cosine starts at its maximum and the maximum temperature occurs at [latex]t = 16[/latex] (4:00 PM), we have [latex]C = 16[/latex].
The temperature at 10:00 AM is 90°F (the average temperature).
The sine function
Sine reaches its maximum [latex]\frac{1}{4}[/latex] period after it crosses the midline going upward. Since our period is 24 hours, [latex]\frac{1}{4}[/latex] period = 6 hours.
The maximum occurs at [latex]t = 16[/latex], so sine crosses the midline going up at: [latex]16 - 6 = 10[/latex]
Choose cosine when your reference point is at a maximum or minimum. Choose sine when your reference point is at the midline. This makes finding the phase shift much easier!