## Modeling with Trigonometric Equations

### Learning Outcomes

• Determine the amplitude and period of sinusoidal functions.
• Model equations and graph sinusoidal functions.
• Model periodic behavior.
• Model harmonic motion functions.

## Determining the Amplitude and Period of a Sinusoidal Function

Any motion that repeats itself in a fixed time period is considered periodic motion and can be modeled by a sinusoidal function. The amplitude of a sinusoidal function is the distance from the midline to the maximum value, or from the midline to the minimum value. The midline is the average value. Sinusoidal functions oscillate above and below the midline, are periodic, and repeat values in set cycles. Recall from Graphs of the Sine and Cosine Functions that the period of the sine function and the cosine function is $\text{ }2\pi .\text{ }$ In other words, for any value of $\text{ }x$,

$\sin \left(x\pm 2\pi k\right)=\sin x\text{ and }\cos \left(x\pm 2\pi k\right)=\cos x\text{ where }k\text{ is an integer}$

### A General Note: Standard Form of Sinusoidal Equations

The general forms of a sinusoidal equation are given as

$y=A\sin \left(Bt-C\right)+D\text{ or }y=A\cos \left(Bt-C\right)+D$

where $\text{amplitude}=|A|,B$ is related to period such that the $\text{ period}=\frac{2\pi }{B},C\text{ }$ is the phase shift such that $\text{ }\frac{C}{B}\text{ }$ denotes the horizontal shift, and $\text{ }D\text{ }$ represents the vertical shift from the graph’s parent graph.

Note that the models are sometimes written as $\text{ }y=a\sin \left(\omega t\pm C\right)+D\text{ }$ or $\text{ }y=a\cos \left(\omega t\pm C\right)+D$, and period is given as $\text{ }\frac{2\pi }{\omega }$.

The difference between the sine and the cosine graphs is that the sine graph begins with the average value of the function and the cosine graph begins with the maximum or minimum value of the function.

### Example 1: Showing How the Properties of a Trigonometric Function Can Transform a Graph

Show the transformation of the graph of $\text{ }y=\sin x\text{ }$ into the graph of $\text{ }y=2\sin \left(4x-\frac{\pi }{2}\right)+2$.

### Example 2: Finding the Amplitude and Period of a Function

Find the amplitude and period of the following functions and graph one cycle.

1. $y=2\sin \left(\frac{1}{4}x\right)$
2. $y=-3\sin \left(2x+\frac{\pi }{2}\right)$
3. $y=\cos x+3$

### Try It

What are the amplitude and period of the function $\text{ }y=3\cos \left(3\pi x\right)?$

## Finding Equations and Graphing Sinusoidal Functions

One method of graphing sinusoidal functions is to find five key points. These points will correspond to intervals of equal length representing $\text{ }\frac{1}{4}\text{ }$ of the period. The key points will indicate the location of maximum and minimum values. If there is no vertical shift, they will also indicate x-intercepts. For example, suppose we want to graph the function $\text{ }y=\cos \theta$. We know that the period is $2\pi$, so we find the interval between key points as follows.

$\frac{2\pi }{4}=\frac{\pi }{2}$

Starting with $\text{ }\theta =0$, we calculate the first y-value, add the length of the interval $\text{ }\frac{\pi }{2}\text{ }$ to 0, and calculate the second y-value. We then add $\text{ }\frac{\pi }{2}\text{ }$ repeatedly until the five key points are determined. The last value should equal the first value, as the calculations cover one full period. Making a table similar to the one below, we can see these key points clearly on the graph shown in Figure 6.

 $\theta$ $0$ $\frac{\pi }{2}$ $\pi$ $\frac{3\pi }{2}$ $2\pi$ $y=\cos \theta$ $1$ $0$ $-1$ $0$ $1$

Figure 6

### Example 3: Graphing Sinusoidal Functions Using Key Points

Graph the function $\text{ }y=-4\cos \left(\pi x\right)\text{ }$ using amplitude, period, and key points.

### Try It

Graph the function $\text{ }y=3\sin \left(3x\right)\text{ }$ using the amplitude, period, and five key points.

## Modeling Periodic Behavior

We will now apply these ideas to problems involving periodic behavior.

### Example 4: Modeling an Equation and Sketching a Sinusoidal Graph to Fit Criteria

The average monthly temperatures for a small town in Oregon are given in the table below. Find a sinusoidal function of the form $y=A\sin \left(Bt-C\right)+D$ that fits the data (round to the nearest tenth) and sketch the graph.

Month Temperature, ${}^{\text{o}}\text{F}$
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

### Example 5: 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 $y$ equal the distance from the tip of the hour hand to the ceiling $x$ hours after noon. Find the equation that models the motion of the clock and sketch the graph.

### Example 6: 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.

### Try It

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

### Example 7: Interpreting the Periodic Behavior Equation

The average person’s blood pressure is modeled by the function $f\left(t\right)=20\sin \left(160\pi t\right)+100$, where $f\left(t\right)$ represents the blood pressure at time $t$, measured in minutes. Interpret the function in terms of period and frequency. Sketch the graph and find the blood pressure reading.

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 $t=0,d=0$.

### A General Note: Simple Harmonic Motion

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

$d=a\cos \left(\omega t\right)\text{ or }d=a\sin \left(\omega t\right)$

where $|a|$ is the amplitude, $\frac{2\pi }{\omega }$ is the period, and $\frac{\omega }{2\pi }$ is the frequency, or the number of cycles per unit of time.

### Example 8: Finding the Displacement, Period, and Frequency, and Graphing a Function

For the given functions,

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

## Damped Harmonic Motion

In reality, a pendulum does not swing back and forth forever, nor does an object on a spring bounce up and down forever. Eventually, the pendulum stops swinging and the object stops bouncing and both return to equilibrium. Periodic motion in which an energy-dissipating force, or damping factor, acts is known as damped harmonic motion. Friction is typically the damping factor.

In physics, various formulas are used to account for the damping factor on the moving object. Some of these are calculus-based formulas that involve derivatives. For our purposes, we will use formulas for basic damped harmonic motion models.

### A General Note: Damped Harmonic Motion

In damped harmonic motion, the displacement of an oscillating object from its rest position at time $t$ is given as

$f\left(t\right)=a{e}^{-ct}\sin \left(\omega t\right)\text{or} f\left(t\right)=a{e}^{-ct}\cos \left(\omega t\right)$

where $c$ is a damping factor, $|a|$ is the initial displacement and $\frac{2\pi }{\omega }$ is the period.

### Example 9: Modeling Damped Harmonic Motion

Model the equations that fit the two scenarios and use a graphing utility to graph the functions: Two mass-spring systems exhibit damped harmonic motion at a frequency of $0.5$ cycles per second. Both have an initial displacement of 10 cm. The first has a damping factor of $0.5$ and the second has a damping factor of $0.1$.

### Example 10: Finding a Cosine Function that Models Damped Harmonic Motion

Find and graph a function of the form $y=a{e}^{-ct}\cos \left(\omega t\right)$ that models the information given.

1. $a=20,c=0.05,p=4$
2. $a=2,c=1.5,f=3$

### Try It

The following equation represents a damped harmonic motion model: $\text{ }f\left(t\right)=5{e}^{-6t}\cos \left(4t\right)$
Find the initial displacement, the damping constant, and the frequency.

### Example 11: Finding a Sine Function that Models Damped Harmonic Motion

Find and graph a function of the form $y=a{e}^{-ct}\sin \left(\omega t\right)$ that models the information given.

1. $a=7,c=10,p=\frac{\pi }{6}$
2. $a=0.3,c=0.2,f=20$

### Try It

Write the equation for damped harmonic motion given $a=10,c=0.5$, and $p=2$.

### Example 12: Modeling the Oscillation of a Spring

A spring measuring 10 inches in natural length is compressed by 5 inches and released. It oscillates once every 3 seconds, and its amplitude decreases by 30% every second. Find an equation that models the position of the spring $t$ seconds after being released.

### Try It

A mass suspended from a spring is raised a distance of 5 cm above its resting position. The mass is released at time $t=0$ and allowed to oscillate. After $\frac{1}{3}$ second, it is observed that the mass returns to its highest position. Find a function to model this motion relative to its initial resting position.

### Example 13: Finding the Value of the Damping Constant c According to the Given Criteria

A guitar string is plucked and vibrates in damped harmonic motion. The string is pulled and displaced 2 cm from its resting position. After 3 seconds, the displacement of the string measures 1 cm. Find the damping constant.

## Bounding Curves in Harmonic Motion

Harmonic motion graphs may be enclosed by bounding curves. When a function has a varying amplitude, such that the amplitude rises and falls multiple times within a period, we can determine the bounding curves from part of the function.

### Example 14: Graphing an Oscillating Cosine Curve

Graph the function $f\left(x\right)=\cos \left(2\pi x\right)\cos \left(16\pi x\right)$.

Key Equations

 Standard form of sinusoidal equation $y=A\sin \left(Bt-C\right)+D\text{or}y=A\cos \left(Bt-C\right)+D$ Simple harmonic motion $d=a\cos \left(\omega t\right)\text{ or }d=a\sin \left(\omega t\right)$ Damped harmonic motion $f\left(t\right)=a{e}^{-c}{}^{t}\sin \left(\omega t\right)\text{or}f\left(t\right)=a{e}^{-ct}\cos \left(\omega t\right)$

## 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.
• Damped harmonic motion is a form of periodic behavior affected by a damping factor. Energy dissipating factors, like friction, cause the displacement of the object to shrink.
• Bounding curves delineate the graph of harmonic motion with variable maximum and minimum values.

## Glossary

damped harmonic motion
oscillating motion that resembles periodic motion and simple harmonic motion, except that the graph is affected by a damping factor, an energy dissipating influence on the motion, such as friction
simple harmonic motion
a repetitive motion that can be modeled by periodic sinusoidal oscillation