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x y
0 5
1 3
2 5
3 13
4 5
5 3
6 5

5 8 sin ( x π 2 )

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x y
3 1 2
2 1
1 1 2
0 0
1 2 1
2 1
3 2 + 1
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x y
1 3 2
0 0
1 2 3
2 3 3
3 1
4 3
5 2 + 3

tan ( x π 12 )

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Graphical

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

f ( x ) = 30 cos ( x π 6 ) 20 cos 2 ( x π 6 ) + 80 [ 0 , 12 ]

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f ( x ) = 18 cos ( x π 12 ) 5 sin ( x π 12 ) + 100 on the interval [ 0 , 24 ]

Graph of f(x) = -18cos(x*pi/12) - 5sin(x*pi/12) + 100 on the interval [0,24]. There is a single peak around 12.

Answers will vary. Sample answer: This function could model temperature changes over the course of one very hot day in Phoenix, Arizona.

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f ( x ) = 10 sin ( x π 6 ) + 24 tan ( x π 240 ) on the interval [ 0 , 80 ]

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Technology

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

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.

9 years from now

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Real-world applications

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

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of 105 °F occurs at 5PM and the average temperature for the day is 85 °F . Find the temperature, to the nearest degree, at 9AM.

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Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of 84 °F occurs at 6PM and the average temperature for the day is 70 °F . Find the temperature, to the nearest degree, at 7AM.

56   ° F

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Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between 47 °F and 63 °F during the day and the average daily temperature first occurs at 10 AM. How many hours after midnight does the temperature first reach 51 °F?

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Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between 64 °F and 86 °F during the day and the average daily temperature first occurs at 12 AM. How many hours after midnight does the temperature first reach 70 °F?

1.8024 hours

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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?

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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

4:30

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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.

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Practice Key Terms 2

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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