Sinusoidal Modeling

Many real-world phenomena follow a predictable, repeating pattern. The rise and fall of tides, the daily cycle of temperatures, and the height of a rider on a Ferris wheel are all examples of periodic behavior. We can model these scenarios mathematically using sinusoidal modeling, which involves fitting a sine or cosine function to the data.

The General Equation

Any sinusoidal model can be written in one of two forms:

$y = A \sin(B(x - C)) + D$ $y = A \cos(B(x - C)) + D$

Each parameter in these equations corresponds to a specific physical feature of the model:

• $A$ (Amplitude): Half the distance between the maximum and minimum values. It represents the peak variation from the average.
• $D$ (Midline / Vertical Shift): The average value of the function. It is the horizontal line halfway between the maximum and minimum.
• $B$ (Frequency parameter): Determines the period of the function. The period $P$ is the time it takes to complete one full cycle, and they are related by the formula $P = \frac{2\pi}{|B|}$.
• $C$ (Phase Shift): The horizontal shift of the graph. It tells you where a cycle "starts" along the x-axis.

How to Find the Parameters

When given data or a word problem, follow these steps to build your model:

1. Find the Midline ($D$): $D = \frac{\text{Maximum} + \text{Minimum}}{2}$
2. Find the Amplitude ($A$): $A = \frac{\text{Maximum} - \text{Minimum}}{2}$
3. Find $B$ from the Period ($P$): $B = \frac{2\pi}{P}$
4. Determine the Phase Shift ($C$): Choose sine or cosine based on your starting point.
   • If you know when the maximum occurs, use positive cosine and let $C$ be that x-value.
   • If you know when the minimum occurs, use negative cosine ($-A \cos(...)$) and let $C$ be that x-value.
   • If you know when the value crosses the midline going up, use positive sine.

Example 1: Temperature Cycle

Problem: The temperature varies between $20^\circ$F and $80^\circ$F over a 24-hour period, peaking at 3 PM. Write a trigonometric model for the temperature $T$ in terms of hours $t$ past midnight.

Solution:

• Midline ($D$): $D = \frac{80 + 20}{2} = 50^\circ$F.
• Amplitude ($A$): $A = \frac{80 - 20}{2} = 30^\circ$F.
• Period ($P$): A full daily cycle takes 24 hours, so $P = 24$. $B = \frac{2\pi}{24} = \frac{\pi}{12}$
• Phase Shift ($C$): The peak occurs at 3 PM, which is $t = 15$ hours past midnight. Since we know the time of the maximum, it is easiest to use a positive cosine function with a phase shift of $C = 15$.

Model: $T(t) = 30 \cos\left(\frac{\pi}{12}(t - 15)\right) + 50$

Example 2: Circular Motion (Ferris Wheel)

Problem: A Ferris wheel has a radius of 30 ft and its center is 35 ft above the ground. It completes one full revolution every 2 minutes. Write a model for a rider's height $h$ in feet over time $t$ in minutes, assuming the rider boards at the very bottom at $t = 0$.

Solution:

• Midline ($D$): The center of the Ferris wheel is 35 ft high, so the average height is $D = 35$.
• Amplitude ($A$): The radius is 30 ft, meaning the rider goes 30 ft above and 30 ft below the center. So, $A = 30$.
• Period ($P$): One revolution takes 2 minutes, so $P = 2$. $B = \frac{2\pi}{2} = \pi$
• Phase Shift ($C$): The rider starts at the bottom (minimum height) at $t = 0$. A negative cosine function starts at a minimum when the input is 0, so we don't need a phase shift ($C = 0$) if we use $-A \cos(...)$.

Model: $h(t) = -30 \cos(\pi t) + 35$