Integrated Math 3 / Unit 5: Trigonometric Functions
5.5 Modeling Periodic Phenomena
Notes
Use sine or cosine to model repeating patterns: tides, temperatures, Ferris wheels, sound waves.
- From the maximum $M$ and minimum $m$: amplitude $\displaystyle = \frac{M - m}{2}$ and midline $\displaystyle = \frac{M + m}{2}$.
- From the period $P$: $\displaystyle B = \frac{2\pi}{P}$.
- Starting at a maximum? Use cosine. Starting at a minimum? Use negative cosine. Starting at the midline going up? Use sine.
- Example: a Ferris wheel of radius 20 m with center 22 m high that turns once every 8 minutes, starting at the bottom: $\displaystyle h(t) = -20\cos\left(\frac{\pi}{4}t\right) + 22$.
Practice
Daily temperatures range from a low of 30 degrees to a high of 90 degrees. What are the amplitude and midline of a sinusoidal model?
Amplitude $\displaystyle = \frac{90 - 30}{2} = 30$ and midline $\displaystyle = \frac{90 + 30}{2} = 60$.
A model for temperature is $\displaystyle T(t) = 15\sin\left(\frac{\pi}{12}(t - 9)\right) + 60$, with $t$ in hours. What is the period?
Period $\displaystyle = \frac{2\pi}{\pi/12} = 24$.
Using $\displaystyle h(t) = -20\cos\left(\frac{\pi}{4}t\right) + 22$ for the height in meters of a Ferris wheel car after $t$ minutes, what is the height at $t = 2$?
$\displaystyle \cos\frac{\pi}{2} = 0$, so $h = 22$. The car is halfway up, at the midline.
Ready for more? Take the Unit 5 test.