# 7.6 Modeling with trigonometric equations  (Page 6/14)

 Page 6 / 14

## Bounding curves in harmonic motion

Harmonic motion graphs may be enclosed by bounding curves. When a function has a varying amplitude    , such that the amplitude rises and falls multiple times within a period, we can determine the bounding curves from part of the function.

## Graphing an oscillating cosine curve

Graph the function $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{cos}\left(2\pi x\right)\mathrm{cos}\left(16\pi x\right).$

The graph produced by this function will be shown in two parts. The first graph will be the exact function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ (see [link] ), and the second graph is the exact function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ plus a bounding function (see [link] . The graphs look quite different.

Access these online resources for additional instruction and practice with trigonometric applications.

Visit this website for additional practice questions from Learningpod.

## Key equations

 Standard form of sinusoidal equation $y=A\text{\hspace{0.17em}}\mathrm{sin}\left(Bt-C\right)+D\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}y=A\text{\hspace{0.17em}}\mathrm{cos}\left(Bt-C\right)+D$ Simple harmonic motion Damped harmonic motion $f\left(t\right)=a{e}^{-c}{}^{t}\mathrm{sin}\left(\omega t\right)\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}f\left(t\right)=a{e}^{-ct}\mathrm{cos}\left(\omega t\right)$

## Key concepts

• Sinusoidal functions are represented by the sine and cosine graphs. In standard form, we can find the amplitude, period, and horizontal and vertical shifts. See [link] and [link] .
• Use key points to graph a sinusoidal function. The five key points include the minimum and maximum values and the midline values. See [link] .
• Periodic functions can model events that reoccur in set cycles, like the phases of the moon, the hands on a clock, and the seasons in a year. See [link] , [link] , [link] and [link] .
• Harmonic motion functions are modeled from given data. Similar to periodic motion applications, harmonic motion requires a restoring force. Examples include gravitational force and spring motion activated by weight. See [link] .
• Damped harmonic motion is a form of periodic behavior affected by a damping factor. Energy dissipating factors, like friction, cause the displacement of the object to shrink. See [link] , [link] , [link] , [link] , and [link] .
• Bounding curves delineate the graph of harmonic motion with variable maximum and minimum values. See [link] .

## Verbal

Explain what types of physical phenomena are best modeled by sinusoidal functions. What are the characteristics necessary?

Physical behavior should be periodic, or cyclical.

What information is necessary to construct a trigonometric model of daily temperature? Give examples of two different sets of information that would enable modeling with an equation.

If we want to model cumulative rainfall over the course of a year, would a sinusoidal function be a good model? Why or why not?

Since cumulative rainfall is always increasing, a sinusoidal function would not be ideal here.

Explain the effect of a damping factor on the graphs of harmonic motion functions.

## Algebraic

For the following exercises, find a possible formula for the trigonometric function represented by the given table of values.

 $x$ $y$ $0$ $-4$ $3$ $-1$ $6$ $2$ $9$ $-1$ $12$ $-4$ $15$ $-1$ $18$ $2$

$y=-3\mathrm{cos}\left(\frac{\pi }{6}x\right)-1$

 $x$ $y$ $0$ $5$ $2$ $1$ $4$ $-3$ $6$ $1$ $8$ $5$ $10$ $1$ $12$ $-3$
 $x$ $y$ $0$ $2$ $\frac{\pi }{4}$ $7$ $\frac{\pi }{2}$ $2$ $\frac{3\pi }{4}$ $-3$ $\pi$ $2$ $\frac{5\pi }{4}$ $7$ $\frac{3\pi }{2}$ $2$

$5\mathrm{sin}\left(2x\right)+2$

 $x$ $y$ $0$ $2$ $\frac{\pi }{4}$ $7$ $\frac{\pi }{2}$ $2$ $\frac{3\pi }{4}$ $-3$ $\pi$ $2$ $\frac{5\pi }{4}$ $7$ $\frac{3\pi }{2}$ $2$
 $x$ $y$ $0$ $1$ $1$ $-3$ $2$ $-7$ $3$ $-3$ $4$ $1$ $5$ $-3$ $6$ $-7$

$4\mathrm{cos}\left(\frac{x\pi }{2}\right)-3$

 $x$ $y$ $0$ $-2$ $1$ $4$ $2$ $10$ $3$ $4$ $4$ $-2$ $5$ $4$ $6$ $10$

can you not take the square root of a negative number
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
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
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
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