Determine the amplitude and period of a periodic context
Model periodic behavior with sinusoidal functions
Write both a sine and cosine function to model the same periodic behavior
Any motion that repeats itself in a fixed time period is considered periodic motion and can be modeled by a sinusoidal function. Sinusoidal functions oscillate above and below the midline, are periodic, and repeat values in set cycles.
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.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],
[latex]\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}[/latex]
The general forms of a sinusoidal equation are given as
[latex]y=A\sin \left(Bt-C\right)+D\text{ or }y=A\cos \left(Bt-C\right)+D[/latex]
where [latex]\text{amplitude}=|A|,B[/latex] is related to period such that the [latex]\text{ period}=\frac{2\pi }{B},C\text{ }[/latex] is the phase shift such that [latex]\text{ }\frac{C}{B}\text{ }[/latex] denotes the horizontal shift, and [latex]\text{ }D\text{ }[/latex] represents the vertical shift from the graph’s parent graph.
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.
Modeling Periodic Behavior
The average monthly temperatures for a small town in Oregon are given in the table below. Find a sinusoidal function of the form [latex]y=A\sin \left(Bt-C\right)+D[/latex] that fits the data (round to the nearest tenth) and sketch the graph.
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
Recall that amplitude is found using the formula
[latex]A=\frac{\text{largest value }-\text{smallest value}}{2}[/latex]
We have the equation [latex]y=13.3\sin \left(\frac{\pi }{6}x-\frac{2\pi }{3}\right)+55.8[/latex]. See the graph in Figure 8.
Figure 8
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.
Begin by making a table of values as shown in the table below.
[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]
To model an equation, we first need to find the amplitude.
There is no horizontal shift, so [latex]C=0[/latex]. Since the function begins with the minimum value of [latex]y[/latex] when [latex]x=0[/latex] (as opposed to the maximum value), we will use the cosine function with the negative value for [latex]A[/latex]. In the form [latex]y=A\cos \left(Bx\pm C\right)+D[/latex], the equation is
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.
As the water level varies from 7 ft to 15 ft, we can calculate the amplitude as
The cycle repeats every 12 hours; therefore, [latex]B[/latex] is
[latex]\frac{2\pi }{12}=\frac{\pi }{6}[/latex]
There is a vertical translation of [latex]\frac{\left(15+8\right)}{2}=11.5[/latex]. Since the value of the function is at a maximum at [latex]t=0[/latex], we will use the cosine function, with the positive value for [latex]A[/latex].
The daily temperature in the month of March in a certain city varies from a low of [latex]24^\circ\text{F}[/latex] to a high of [latex]40^\circ\text{F}[/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^\circ\text{F}[/latex]. Let [latex]t=0[/latex] correspond to noon.
[latex]y=8\sin \left(\frac{\pi }{12}t\right)+32[/latex]
The temperature reaches freezing at noon and at midnight.
