# 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.

find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what