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

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

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

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

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

## 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.

If the plane intersects the cone (either above or below) horizontally, what figure will be created?
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
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
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
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
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