# 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

the sum of any two linear polynomial is what
divide simplify each answer 3/2÷5/4
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard