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 [latex]\text{ }2\pi .\text{ }[/latex] In other words, for any value of [latex]\text{ }x[/latex],

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 [latex]\text{ }\frac{1}{4}\text{ }[/latex] 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 [latex]\text{ }y=\cos \theta[/latex]. We know that the period is [latex]2\pi[/latex], so we find the interval between key points as follows.

Starting with [latex]\text{ }\theta =0[/latex], we calculate the first y-value, add the length of the interval [latex]\text{ }\frac{\pi }{2}\text{ }[/latex] to 0, and calculate the second y-value. We then add [latex]\text{ }\frac{\pi }{2}\text{ }[/latex] 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.

[latex]\theta[/latex]	[latex]0[/latex]	[latex]\frac{\pi }{2}[/latex]	[latex]\pi[/latex]	[latex]\frac{3\pi }{2}[/latex]	[latex]2\pi[/latex]
[latex]y=\cos \theta[/latex]	[latex]1[/latex]	[latex]0[/latex]	[latex]-1[/latex]	[latex]0[/latex]	[latex]1[/latex]

Modeling Periodic Behavior

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

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].

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.

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.

Standard form of sinusoidal equation	[latex]y=A\sin \left(Bt-C\right)+D\text{or}y=A\cos \left(Bt-C\right)+D[/latex]
Simple harmonic motion	[latex]d=a\cos \left(\omega t\right)\text{ or }d=a\sin \left(\omega t\right)[/latex]
Damped harmonic motion	[latex]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)[/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.
• 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.

 

Section 7.8 Homework Exercises

1. Explain what types of physical phenomena are best modeled by sinusoidal functions. What are the characteristics necessary?

2. What information is necessary to construct a trigonometric model of daily temperature? Give examples of two different sets of information that would enable modeling with an equation.

3. If we want to model cumulative rainfall over the course of a year, would a sinusoidal function be a good model? Why or why not?

4. Explain the effect of a damping factor on the graphs of harmonic motion functions.

For the following exercises, find a possible formula for the trigonometric function represented by the given table of values.

5.

[latex]x[/latex]	[latex]y[/latex]
[latex]0[/latex]	[latex]-4[/latex]
[latex]3[/latex]	[latex]-1[/latex]
[latex]6[/latex]	[latex]2[/latex]
[latex]9[/latex]	[latex]-1[/latex]
[latex]12[/latex]	[latex]-4[/latex]
[latex]15[/latex]	[latex]-1[/latex]
[latex]18[/latex]	[latex]2[/latex]

6.

[latex]x[/latex]	[latex]y[/latex]
[latex]0[/latex]	[latex]5[/latex]
[latex]2[/latex]	[latex]1[/latex]
[latex]4[/latex]	[latex]-3[/latex]
[latex]6[/latex]	[latex]1[/latex]
[latex]8[/latex]	[latex]5[/latex]
[latex]10[/latex]	[latex]1[/latex]
[latex]12[/latex]	[latex]-3[/latex]

7.

[latex]x[/latex]	[latex]y[/latex]
[latex]0[/latex]	[latex]2[/latex]
[latex]\frac{\pi }{4}[/latex]	[latex]7[/latex]
[latex]\frac{\pi }{2}[/latex]	[latex]2[/latex]
[latex]\frac{3\pi }{4}[/latex]	[latex]-3[/latex]
[latex]\pi[/latex]	[latex]2[/latex]
[latex]\frac{5\pi }{4}[/latex]	[latex]7[/latex]
[latex]\frac{3\pi }{2}[/latex]	[latex]2[/latex]

8.

[latex]x[/latex]	[latex]y[/latex]
[latex]0[/latex]	[latex]2[/latex]
[latex]\frac{\pi }{4}[/latex]	[latex]7[/latex]
[latex]\frac{\pi }{2}[/latex]	[latex]2[/latex]
[latex]\frac{3\pi }{4}[/latex]	[latex]-3[/latex]
[latex]\pi[/latex]	[latex]2[/latex]
[latex]\frac{5\pi }{4}[/latex]	[latex]7[/latex]
[latex]\frac{3\pi }{2}[/latex]	[latex]2[/latex]

9.

[latex]x[/latex]	[latex]y[/latex]
[latex]0[/latex]	[latex]1[/latex]
[latex]1[/latex]	[latex]-3[/latex]
[latex]2[/latex]	[latex]-7[/latex]
[latex]3[/latex]	[latex]-3[/latex]
[latex]4[/latex]	[latex]1[/latex]
[latex]5[/latex]	[latex]-3[/latex]
[latex]6[/latex]	[latex]-7[/latex]

10.

[latex]x[/latex]	[latex]y[/latex]
[latex]0[/latex]	[latex]-2[/latex]
[latex]1[/latex]	[latex]4[/latex]
[latex]2[/latex]	[latex]10[/latex]
[latex]3[/latex]	[latex]4[/latex]
[latex]4[/latex]	[latex]-2[/latex]
[latex]5[/latex]	[latex]4[/latex]
[latex]6[/latex]	[latex]10[/latex]

11.

[latex]x[/latex]	[latex]y[/latex]
[latex]0[/latex]	[latex]5[/latex]
[latex]1[/latex]	[latex]-3[/latex]
[latex]2[/latex]	[latex]5[/latex]
[latex]3[/latex]	[latex]13[/latex]
[latex]4[/latex]	[latex]5[/latex]
[latex]5[/latex]	[latex]-3[/latex]
[latex]6[/latex]	[latex]5[/latex]

12.

[latex]x[/latex]	[latex]y[/latex]
[latex]-3[/latex]	[latex]-1-\sqrt{2}[/latex]
[latex]-2[/latex]	[latex]-1[/latex]
[latex]-1[/latex]	[latex]1-\sqrt{2}[/latex]
[latex]0[/latex]	[latex]0[/latex]
[latex]1[/latex]	[latex]\sqrt{2}-1[/latex]
[latex]2[/latex]	[latex]1[/latex]
[latex]3[/latex]	[latex]\sqrt{2}+1[/latex]

13.

[latex]x[/latex]	[latex]y[/latex]
[latex]-1[/latex]	[latex]\sqrt{3}-2[/latex]
[latex]0[/latex]	[latex]0[/latex]
[latex]1[/latex]	[latex]2-\sqrt{3}[/latex]
[latex]2[/latex]	[latex]\frac{\sqrt{3}}{3}[/latex]
[latex]3[/latex]	[latex]1[/latex]
[latex]4[/latex]	[latex]\sqrt{3}[/latex]
[latex]5[/latex]	[latex]2+\sqrt{3}[/latex]

For the following exercises, graph the given function, and then find a possible physical process that the equation could model.

14. [latex]f\left(x\right)=-30\cos \left(\frac{x\pi }{6}\right)-20{\cos }^{2}\left(\frac{x\pi }{6}\right)+80\left[0,12\right][/latex]

15. [latex]f\left(x\right)=-18\cos \left(\frac{x\pi }{12}\right)-5\sin \left(\frac{x\pi }{12}\right)+100[/latex] on the interval [latex]\left[0,24\right][/latex]

16. [latex]f\left(x\right)=10-\sin \left(\frac{x\pi }{6}\right)+24\tan \left(\frac{x\pi }{240}\right)[/latex] on the interval [latex]\left[0,80\right][/latex]

For the following exercise, construct a function modeling behavior and use a calculator to find desired results.

17. A city’s average yearly rainfall is currently 20 inches and varies seasonally by 5 inches. Due to unforeseen circumstances, rainfall appears to be decreasing by 15% each year. How many years from now would we expect rainfall to initially reach 0 inches? Note, the model is invalid once it predicts negative rainfall, so choose the first point at which it goes below 0.

For the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.

18. Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of [latex]105\text{^\circ F}[/latex] occurs at 5PM and the average temperature for the day is [latex]85\text{^\circ F}\text{.}[/latex] Find the temperature, to the nearest degree, at 9AM.

19. Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of [latex]84\text{^\circ F}[/latex] occurs at 6PM and the average temperature for the day is [latex]70\text{^\circ F}\text{.}[/latex] Find the temperature, to the nearest degree, at 7AM.

20. Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between [latex]47\text{^\circ F}[/latex] and [latex]63\text{^\circ F}[/latex] during the day and the average daily temperature first occurs at 10 AM. How many hours after midnight does the temperature first reach [latex]51\text{^\circ F?}[/latex]

21. Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between [latex]64^\circ\text{F}[/latex] and [latex]86^\circ\text{F}[/latex] during the day and the average daily temperature first occurs at 12 AM. How many hours after midnight does the temperature first reach [latex]70^\circ\text{F?}[/latex]

22. A Ferris wheel is 20 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 6 minutes. How much of the ride, in minutes and seconds, is spent higher than 13 meters above the ground?

23. A Ferris wheel is 45 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. How many minutes of the ride are spent higher than 27 meters above the ground? Round to the nearest second

24. The sea ice area around the North Pole fluctuates between about 6 million square kilometers on September 1 to 14 million square kilometers on March 1. Assuming a sinusoidal fluctuation, when are there less than 9 million square kilometers of sea ice? Give your answer as a range of dates, to the nearest day.

25. The sea ice area around the South Pole fluctuates between about 18 million square kilometers in September to 3 million square kilometers in March. Assuming a sinusoidal fluctuation, when are there more than 15 million square kilometers of sea ice? Give your answer as a range of dates, to the nearest day.

26. During a 90-day monsoon season, daily rainfall can be modeled by sinusoidal functions. If the rainfall fluctuates between a low of 2 inches on day 10 and 12 inches on day 55, during what period is daily rainfall more than 10 inches?

27. During a 90-day monsoon season, daily rainfall can be modeled by sinusoidal functions. A low of 4 inches of rainfall was recorded on day 30, and overall the average daily rainfall was 8 inches. During what period was daily rainfall less than 5 inches?

28. In a certain region, monthly precipitation peaks at 8 inches on June 1 and falls to a low of 1 inch on December 1. Identify the periods when the region is under flood conditions (greater than 7 inches) and drought conditions (less than 2 inches). Give your answer in terms of the nearest day.

29. In a certain region, monthly precipitation peaks at 24 inches in September and falls to a low of 4 inches in March. Identify the periods when the region is under flood conditions (greater than 22 inches) and drought conditions (less than 5 inches). Give your answer in terms of the nearest day.

For the following exercises, find the amplitude, period, and frequency of the given function.

30. The displacement [latex]h\left(t\right)[/latex] in centimeters of a mass suspended by a spring is modeled by the function [latex]h\left(t\right)=8\sin \left(6\pi t\right)[/latex], where [latex]t[/latex] is measured in seconds. Find the amplitude, period, and frequency of this displacement.

31. The displacement [latex]h\left(t\right)[/latex] in centimeters of a mass suspended by a spring is modeled by the function [latex]h\left(t\right)=11\sin \left(12\pi t\right)[/latex], where [latex]t[/latex] is measured in seconds. Find the amplitude, period, and frequency of this displacement.

32. The displacement [latex]h\left(t\right)[/latex] in centimeters of a mass suspended by a spring is modeled by the function [latex]h\left(t\right)=4\cos \left(\frac{\pi }{2}t\right)[/latex], where [latex]t[/latex] is measured in seconds. Find the amplitude, period, and frequency of this displacement.

For the following exercises, construct an equation that models the described behavior.

33. The displacement [latex]h\left(t\right)[/latex], in centimeters, of a mass suspended by a spring is modeled by the function [latex]h\left(t\right)=-5\cos \left(60\pi t\right)[/latex], where [latex]t[/latex] is measured in seconds. Find the amplitude, period, and frequency of this displacement.

For the following exercises, construct an equation that models the described behavior.

34. A deer population oscillates 19 above and below average during the year, reaching the lowest value in January. The average population starts at 800 deer and increases by 160 each year. Find a function that models the population, [latex]P[/latex], in terms of months since January, [latex]t[/latex].

35. A rabbit population oscillates 15 above and below average during the year, reaching the lowest value in January. The average population starts at 650 rabbits and increases by 110 each year. Find a function that models the population, [latex]P[/latex], in terms of months since January, [latex]t[/latex].

36. A muskrat population oscillates 33 above and below average during the year, reaching the lowest value in January. The average population starts at 900 muskrats and increases by 7% each month. Find a function that models the population, [latex]P[/latex], in terms of months since January, [latex]t[/latex].

37. A fish population oscillates 40 above and below average during the year, reaching the lowest value in January. The average population starts at 800 fish and increases by 4% each month. Find a function that models the population, [latex]P[/latex], in terms of months since January, [latex]t[/latex].

38. A spring attached to the ceiling is pulled 10 cm down from equilibrium and released. The amplitude decreases by 15% each second. The spring oscillates 18 times each second. Find a function that models the distance, [latex]D[/latex], the end of the spring is from equilibrium in terms of seconds, [latex]t[/latex], since the spring was released.

39. A spring attached to the ceiling is pulled 7 cm down from equilibrium and released. The amplitude decreases by 11% each second. The spring oscillates 20 times each second. Find a function that models the distance, [latex]D[/latex], the end of the spring is from equilibrium in terms of seconds, [latex]t[/latex], since the spring was released.

40. A spring attached to the ceiling is pulled 17 cm down from equilibrium and released. After 3 seconds, the amplitude has decreased to 13 cm. The spring oscillates 14 times each second. Find a function that models the distance, [latex]D[/latex], the end of the spring is from equilibrium in terms of seconds, [latex]t[/latex], since the spring was released.

41. A spring attached to the ceiling is pulled 19 cm down from equilibrium and released. After 4 seconds, the amplitude has decreased to 14 cm. The spring oscillates 13 times each second. Find a function that models the distance, [latex]D[/latex], the end of the spring is from equilibrium in terms of seconds, [latex]t[/latex], since the spring was released.

For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.

42. A certain lake currently has an average trout population of 20,000. The population naturally oscillates above and below average by 2,000 every year. This year, the lake was opened to fishermen. If fishermen catch 3,000 fish every year, how long will it take for the lake to have no more trout?

43. Whitefish populations are currently at 500 in a lake. The population naturally oscillates above and below by 25 each year. If humans overfish, taking 4% of the population every year, in how many years will the lake first have fewer than 200 whitefish?

44. A spring attached to a ceiling is pulled down 11 cm from equilibrium and released. After 2 seconds, the amplitude has decreased to 6 cm. The spring oscillates 8 times each second. Find when the spring first comes between [latex]-0.1[/latex] and [latex]0.1\text{ cm,}[/latex] effectively at rest.

45. A spring attached to a ceiling is pulled down 21 cm from equilibrium and released. After 6 seconds, the amplitude has decreased to 4 cm. The spring oscillates 20 times each second. Find when the spring first comes between [latex]-0.1[/latex] and [latex]0.1\text{ cm,}[/latex] effectively at rest.

46. Two springs are pulled down from the ceiling and released at the same time. The first spring, which oscillates 8 times per second, was initially pulled down 32 cm from equilibrium, and the amplitude decreases by 50% each second. The second spring, oscillating 18 times per second, was initially pulled down 15 cm from equilibrium and after 4 seconds has an amplitude of 2 cm. Which spring comes to rest first, and at what time? Consider “rest” as an amplitude less than [latex]0.1\text{ cm}\text{.}[/latex]

47. Two springs are pulled down from the ceiling and released at the same time. The first spring, which oscillates 14 times per second, was initially pulled down 2 cm from equilibrium, and the amplitude decreases by 8% each second. The second spring, oscillating 22 times per second, was initially pulled down 10 cm from equilibrium and after 3 seconds has an amplitude of 2 cm. Which spring comes to rest first, and at what time? Consider “rest” as an amplitude less than [latex]0.1\text{ cm}\text{.}[/latex]

48. A plane flies 1 hour at 150 mph at [latex]{22}^{\circ }[/latex] east of north, then continues to fly for 1.5 hours at 120 mph, this time at a bearing of [latex]{112}^{\circ }[/latex] east of north. Find the total distance from the starting point and the direct angle flown north of east.

49. A plane flies 2 hours at 200 mph at a bearing of [latex]\text{ }{60}^{\circ }[/latex], then continues to fly for 1.5 hours at the same speed, this time at a bearing of [latex]{150}^{\circ }[/latex]. Find the distance from the starting point and the bearing from the starting point. Hint: bearing is measured counterclockwise from north.

For the following exercises, find a function of the form [latex]y=a{b}^{x}+c\sin \left(\frac{\pi }{2}x\right)[/latex] that fits the given data.

50.

[latex]x[/latex]	0	1	2	3
[latex]y[/latex]	6	29	96	379

51.

[latex]x[/latex]	0	1	2	3
[latex]y[/latex]	6	34	150	746

52.

[latex]x[/latex]	0	1	2	3
[latex]y[/latex]	4	0	16	-40

For the following exercises, find a function of the form [latex]y=a{b}^{x}\cos \left(\frac{\pi }{2}x\right)+c[/latex] that fits the given data.

53.

[latex]x[/latex]	0	1	2	3
[latex]y[/latex]	11	3	1	3

54.

[latex]x[/latex]	0	1	2	3
[latex]y[/latex]	4	1	−11	1