Modeling with Trigonometric Equations: Apply It

  • 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.

  1. 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.
  2. What is the temperature at 10:00 AM?
  3. Write an equivalent sine function to model the same temperature pattern.

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!

The water depth at a dock varies with the tides. High tide of 12 feet occurs at noon, and low tide of 4 feet occurs at 6:00 PM. The cycle repeats every 12 hours.

  1. Find a cosine function [latex]d(t) = A\cos(B(t - C)) + D[/latex] to model the depth, where [latex]t[/latex] is hours after midnight
  2. What is the water depth at 3:00 PM?
  3. Write an equivalent sine function

A weight on a spring oscillates vertically with position given by [latex]y = 7\cos\left(\frac{\pi}{3}t\right)[/latex] cm, where [latex]t[/latex] is time in seconds. Find:

  1. The maximum displacement of the weight
  2. The period of oscillation
  3. The frequency
  4. At what time during the first period does the weight first reach [latex]y = 3.5[/latex] cm.