# 7.6 Modeling with trigonometric equations  (Page 7/14)

 Page 7 / 14
 $x$ $y$ $0$ $5$ $1$ $-3$ $2$ $5$ $3$ $13$ $4$ $5$ $5$ $-3$ $6$ $5$

$5-8\mathrm{sin}\left(\frac{x\pi }{2}\right)$

 $x$ $y$ $-3$ $-1-\sqrt{2}$ $-2$ $-1$ $-1$ $1-\sqrt{2}$ $0$ $0$ $1$ $\sqrt{2}-1$ $2$ $1$ $3$ $\sqrt{2}+1$
 $x$ $y$ $-1$ $\sqrt{3}-2$ $0$ $0$ $1$ $2-\sqrt{3}$ $2$ $\frac{\sqrt{3}}{3}$ $3$ $1$ $4$ $\sqrt{3}$ $5$ $2+\sqrt{3}$

$\mathrm{tan}\left(\frac{x\pi }{12}\right)$

## Graphical

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

$f\left(x\right)=-30\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{x\pi }{6}\right)-20\text{\hspace{0.17em}}{\mathrm{cos}}^{2}\left(\frac{x\pi }{6}\right)+80\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[0,12\right]$

$f\left(x\right)=-18\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{x\pi }{12}\right)-5\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{x\pi }{12}\right)+100\text{\hspace{0.17em}}$ on the interval $\text{\hspace{0.17em}}\left[0,24\right]$ Answers will vary. Sample answer: This function could model temperature changes over the course of one very hot day in Phoenix, Arizona.

$f\left(x\right)=10-\mathrm{sin}\left(\frac{x\pi }{6}\right)+24\text{\hspace{0.17em}}\mathrm{tan}\left(\frac{x\pi }{240}\right)\text{\hspace{0.17em}}$ on the interval $\text{\hspace{0.17em}}\left[0,80\right]$

## 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

## 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 $\text{\hspace{0.17em}}105\text{°F}\text{\hspace{0.17em}}$ occurs at 5PM and the average temperature for the day is $\text{\hspace{0.17em}}85\text{°F}\text{.}\text{\hspace{0.17em}}$ Find the temperature, to the nearest degree, at 9AM.

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of $\text{\hspace{0.17em}}84\text{°F}\text{\hspace{0.17em}}$ occurs at 6PM and the average temperature for the day is $\text{\hspace{0.17em}}70\text{°F}\text{.}\text{\hspace{0.17em}}$ Find the temperature, to the nearest degree, at 7AM.

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between $\text{\hspace{0.17em}}47\text{°F}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}63\text{°F}\text{\hspace{0.17em}}$ during the day and the average daily temperature first occurs at 10 AM. How many hours after midnight does the temperature first reach $\text{\hspace{0.17em}}51\text{°F?}$

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between $\text{\hspace{0.17em}}64\text{°F}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}86\text{°F}\text{\hspace{0.17em}}$ during the day and the average daily temperature first occurs at 12 AM. How many hours after midnight does the temperature first reach $\text{\hspace{0.17em}}70\text{°F?}$

$\text{\hspace{0.17em}}1.8024\text{\hspace{0.17em}}$ hours

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?

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

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.

#### Questions & Answers

how fast can i understand functions without much difficulty
Joe Reply
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply

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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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