3.9 Modeling with Trigonometric Equations

Learning Objectives

In this section, you will:

Model equations and graph sinusoidal functions.
Model periodic behavior.
Model simple harmonic motion functions.

The hands on a clock are periodic: they repeat positions every twelve hours. (credit: “zoutedrop”/Flickr)

Suppose we charted the average daily temperatures in New York City over the course of one year. We would expect to find the lowest temperatures in January and February and highest in July and August. This familiar cycle repeats year after year, and if we were to extend the graph over multiple years, it would resemble a periodic function.

Many other natural phenomena are also periodic. For example, the phases of the moon have a period of approximately 28 days, and birds know to fly south at about the same time each year.

So how can we model an equation to reflect periodic behavior? First, we must collect and record data. We then find a function that resembles an observed pattern. Finally, we make the necessary alterations to the function to get a model that is dependable. In this section, we will take a deeper look at specific types of periodic behavior and model equations to fit data.

Modeling Periodic Behavior

Example 1: Modeling an Equation and Sketching a Sinusoidal Graph to Fit Criteria

The average monthly temperatures for a small town in Oregon are given. Find a sinusoidal function of the form [latex]y=A\mathrm{sin}\left(B\left(t-h\right)\right)+k[/latex] that fits the data (round to the nearest tenth) and sketch the graph.

Table 1
Month	Temperature,[latex]{}^{\text{o}}\text{F}[/latex]
January	42.5
February	44.5
March	48.5
April	52.5
May	58
June	63
July	68.5
August	69
September	64.5
October	55.5
November	46.5
December	43.5

Show Solution

Recall that amplitude is found using the formula

[latex]A=\frac{\text{largest value }-\text{smallest value}}{2}.[/latex][latex]\\[/latex]

Thus, the amplitude is

[latex]\begin{align*} |A|&=\frac{69-42.5}{2}\\ &=13.25.\end{align*}[/latex][latex]\\[/latex]

The data covers a period of 12 months, so [latex]\frac{2\pi }{B}=12[/latex] which gives [latex]B=\frac{2\pi }{12}=\frac{\pi }{6}.[/latex][latex]\\[/latex]

The vertical shift is found using the following equation.[latex]\\[/latex]

[latex]k=\frac{\text{highest value}+\text{lowest value}}{2}.[/latex]

Thus, the vertical shift is[latex]\\[/latex]

[latex]\begin{align*}k&=\frac{69+42.5}{2}\\ &=55.8.\end{align*}[/latex][latex]\\[/latex]

So far, we have the equation [latex]y=13.3\mathrm{sin}\left(\frac{\pi}{6}\left(x-h\right)\right)+55.8.[/latex][latex]\\[/latex]

To find the horizontal shift, we can input the [latex]x[/latex] and [latex]y[/latex] values for the first month, which will be [latex]x=1[/latex] and [latex]y=42.5[/latex]. We can then solve for [latex]h[/latex] as shown below.[latex]\\[/latex]

[latex]\begin{align*}42.5&=13.3\mathrm{sin}\left(\frac{\pi}{6}\left(1-h\right)\right)+55.8\\ -13.3&=13.3\mathrm{sin}\left(\frac{\pi }{6}\left(1-h\right)\right)&&\text{Subtracted 55.8 from both sides. }\\-1&=\mathrm{sin}\left(\frac{\pi} {6}\left(1-h\right)\right) &&\text{Now divide both sides by 13.3.}\end{align*}[/latex][latex]\\[/latex]

We can use the following idea: [latex]\mathrm{sin}\left(\theta\right) =-1\to \theta =-\frac{\pi }{2}[/latex][latex]\\[/latex]

[latex]\begin{align*}\frac{\pi} {6}\left(1-h\right)&=-\frac{\pi }{2}&&\text{Set the input expression of sine equal to }-\frac{\pi}{2}. \\ \frac{\pi}{6}-\frac{\pi}{6}h&=-\frac{\pi}{2}&&\text{Distribute on the left hand side.}\\-\frac{\pi}{6}h&=-\frac{\pi }{2}-\frac{\pi}{6}&&\text{Subtract }\frac{\pi}{6}\text{ from both sides.}\\h&=\left(-\frac{\pi }{2}-\frac{\pi}{6}\right)\frac{-6}{\pi}&&\text{Multiply both sides by }-\frac{6}{\pi}.\\h&=3+1=4 \end{align*}[/latex][latex]\\[/latex]

We have the equation [latex]y=13.3\mathrm{sin}\left(\frac{\pi }{6}\left(x-4\right)\right)+55.8.[/latex] See the graph in Figure 1.

Graph of the equation y=13.3sin(pi/6 x - 2pi/3) + 55.8. The average value is a dotted horizontal line y=55.8, and the amplitude is 13.3

Figure 1

Example 2: Describing Periodic Motion

The hour hand of the large clock on the wall in Union Station measures 24 inches in length. At noon, the tip of the hour hand is 30 inches from the ceiling. At 3 PM, the tip is 54 inches from the ceiling, and at 6 PM, 78 inches. At 9 PM, it is again 54 inches from the ceiling, and at midnight, the tip of the hour hand returns to its original position 30 inches from the ceiling. Let [latex]y[/latex] equal the distance from the tip of the hour hand to the ceiling [latex]x[/latex] hours after noon. Find the equation that models the motion of the clock and sketch the graph.

Show Solution

Table 2
[latex]x[/latex]	[latex]y[/latex]	Points to plot
Noon	30 in	[latex]\left(0,30\right)[/latex]
3 PM	54 in	[latex]\left(3,54\right)[/latex]
6 PM	78 in	[latex]\left(6,78\right)[/latex]
9 PM	54 in	[latex]\left(9,54\right)[/latex]
Midnight	30 in	[latex]\left(12,30\right)[/latex]

Example 3: Determining a Model for Tides

The height of the tide in a small beach town is measured along a seawall. Water levels oscillate between 7 feet at low tide and 15 feet at high tide. On a particular day, low tide occurred at 6 AM and high tide occurred at noon. Approximately every 12 hours, the cycle repeats. Find an equation to model the water levels.

Show Solution

Try it #1

The daily temperature in the month of March in a certain city varies from a low of [latex]24\text{°F}[/latex] to a high of [latex]40\text{°F}\text{.}[/latex] Find a sinusoidal function to model daily temperature and sketch the graph. Approximate the time when the temperature reaches the freezing point [latex]32\text{°F}\text{.}[/latex] Let [latex]t=0[/latex] correspond to noon.

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Example 4: Interpreting the Periodic Behavior Equation

The average person’s blood pressure is modeled by the function [latex]f\left(t\right)=20\mathrm{sin}\left(160\pi t\right)+100,[/latex] where [latex]f\left(t\right)[/latex] represents the blood pressure at time [latex]t,[/latex] measured in minutes. Sketch the graph and find the blood pressure reading.

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Modeling Harmonic Motion Functions

Harmonic motion is a form of periodic motion, but there are factors to consider that differentiate the two types. While general periodic motion applications cycle through their periods with no outside interference, harmonic motion requires a restoring force. Examples of harmonic motion include springs, gravitational force, and magnetic force.

Simple Harmonic Motion

A type of motion described as simple harmonic motion involves a restoring force but assumes that the motion will continue forever. Imagine a weighted object hanging on a spring, When that object is not disturbed, we say that the object is at rest, or in equilibrium. If the object is pulled down and then released, the force of the spring pulls the object back toward equilibrium and harmonic motion begins. The restoring force is directly proportional to the displacement of the object from its equilibrium point. When [latex]t=0,d=0.[/latex]

Simple Harmonic Motion

We see that simple harmonic motion equations are given in terms of displacement:

[latex]d=A\mathrm{cos}\left(Bt\right)\text{ or }d=A\mathrm{sin}\left(Bt\right)[/latex][latex]\\[/latex]

where [latex]|A|[/latex] is the amplitude, and [latex]\frac{2\pi }{B }[/latex] is the period.

Example 5: Finding the Displacement, Period, and Frequency, and Graphing a Function

For the given functions,

Find the maximum displacement of an object.
Find the period or the time required for one vibration.
Sketch the graph.
1. [latex]y=5\mathrm{sin}\left(3t\right)[/latex]
2. [latex]y=6\mathrm{cos}\left(\pi t\right)[/latex]
3. [latex]y=5\mathrm{cos}\left(\frac{\pi }{2}t\right)[/latex]

Show Solution

Access these online resources for additional instruction and practice with trigonometric applications.

Key Equations


Standard form of sinusoidal equation	[latex]y=A\mathrm{sin}\left(B\left(t-h\right)\right)+k\text{ or }y=A\mathrm{cos}\left(B\left(t-h\right)\right)+k[/latex]
Simple harmonic motion	[latex]d=a\mathrm{cos}\left(B t\right)\text{ or }d=a\mathrm{sin}\left(B t\right)[/latex]

Key Concepts

Sinusoidal functions are represented by the sine and cosine graphs. In standard form, we can find the amplitude, period, and horizontal and vertical shifts.
Use key points to graph a sinusoidal function. The five key points include the minimum and maximum values and the midline values.
Periodic functions can model events that reoccur in set cycles, like the phases of the moon, the hands on a clock, and the seasons in a year.
Harmonic motion functions are modeled from given data. Similar to periodic motion applications, harmonic motion requires a restoring force. Examples include gravitational force and spring motion activated by weight.

Glossary

simple harmonic motion: a repetitive motion that can be modeled by periodic sinusoidal oscillation

Trigonometric Functions