# 6.1 Graphs of the sine and cosine functions  (Page 8/13)

 Page 8 / 13

For the equation $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}\mathrm{cos}\left(Bx+C\right)+D,$ what constants affect the range of the function and how do they affect the range?

The absolute value of the constant $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ (amplitude) increases the total range and the constant $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ (vertical shift) shifts the graph vertically.

How does the range of a translated sine function relate to the equation $\text{\hspace{0.17em}}y=A\text{\hspace{0.17em}}\mathrm{sin}\left(Bx+C\right)+D?$

How can the unit circle be used to construct the graph of $\text{\hspace{0.17em}}f\left(t\right)=\mathrm{sin}\text{\hspace{0.17em}}t?$

At the point where the terminal side of $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ intersects the unit circle, you can determine that the $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ equals the y -coordinate of the point.

## Graphical

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y -values and their corresponding x -values on one period for $\text{\hspace{0.17em}}x>0.\text{\hspace{0.17em}}$ Round answers to two decimal places if necessary.

$f\left(x\right)=2\mathrm{sin}\text{\hspace{0.17em}}x$

$f\left(x\right)=\frac{2}{3}\mathrm{cos}\text{\hspace{0.17em}}x$

amplitude: $\text{\hspace{0.17em}}\frac{2}{3};\text{\hspace{0.17em}}$ period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=\frac{2}{3}\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=0;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=-\frac{2}{3}\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=\pi ;\text{\hspace{0.17em}}$ for one period, the graph starts at 0 and ends at $\text{\hspace{0.17em}}2\pi$

$f\left(x\right)=-3\mathrm{sin}\text{\hspace{0.17em}}x$

$f\left(x\right)=4\mathrm{sin}\text{\hspace{0.17em}}x$

amplitude: 4; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ maximum $\text{\hspace{0.17em}}y=4\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=\frac{\pi }{2};\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=-4\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=\frac{3\pi }{2};\text{\hspace{0.17em}}$ one full period occurs from $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}x=2\pi$

$f\left(x\right)=2\mathrm{cos}\text{\hspace{0.17em}}x$

$f\left(x\right)=\mathrm{cos}\left(2x\right)$

amplitude: 1; period: $\text{\hspace{0.17em}}\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=1\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=\pi ;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=-1\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=\frac{\pi }{2};\text{\hspace{0.17em}}$ one full period is graphed from $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}x=\pi$

$f\left(x\right)=2\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{1}{2}x\right)$

$f\left(x\right)=4\text{\hspace{0.17em}}\mathrm{cos}\left(\pi x\right)$

amplitude: 4; period: 2; midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=4\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=0;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=-4\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=1$

$f\left(x\right)=3\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{6}{5}x\right)$

$y=3\text{\hspace{0.17em}}\mathrm{sin}\left(8\left(x+4\right)\right)+5$

amplitude: 3; period: $\text{\hspace{0.17em}}\frac{\pi }{4};\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=5;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=8\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=0.12;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=2\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=0.516;\text{\hspace{0.17em}}$ horizontal shift: $\text{\hspace{0.17em}}-4;\text{\hspace{0.17em}}$ vertical translation 5; one period occurs from $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}x=\frac{\pi }{4}$

$y=2\text{\hspace{0.17em}}\mathrm{sin}\left(3x-21\right)+4$

$y=5\text{\hspace{0.17em}}\mathrm{sin}\left(5x+20\right)-2$

amplitude: 5; period: $\text{\hspace{0.17em}}\frac{2\pi }{5};\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=-2;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=3\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=0.08;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=-7\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=0.71;\text{\hspace{0.17em}}$ phase shift: $\text{\hspace{0.17em}}-4;\text{\hspace{0.17em}}$ vertical translation: $\text{\hspace{0.17em}}-2;\text{\hspace{0.17em}}$ one full period can be graphed on $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}x=\frac{2\pi }{5}$

For the following exercises, graph one full period of each function, starting at $\text{\hspace{0.17em}}x=0.\text{\hspace{0.17em}}$ For each function, state the amplitude, period, and midline. State the maximum and minimum y -values and their corresponding x -values on one period for $\text{\hspace{0.17em}}x>0.\text{\hspace{0.17em}}$ State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

$f\left(t\right)=2\mathrm{sin}\left(t-\frac{5\pi }{6}\right)$

$f\left(t\right)=-\mathrm{cos}\left(t+\frac{\pi }{3}\right)+1$

amplitude: 1 ; period: $\text{\hspace{0.17em}}2\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=1;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=2\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}x=2.09;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=2\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}t=2.09;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=0\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}t=5.24;\text{\hspace{0.17em}}$ phase shift: $\text{\hspace{0.17em}}-\frac{\pi }{3};\text{\hspace{0.17em}}$ vertical translation: 1; one full period is from $\text{\hspace{0.17em}}t=0\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}t=2\pi$

$f\left(t\right)=4\mathrm{cos}\left(2\left(t+\frac{\pi }{4}\right)\right)-3$

$f\left(t\right)=-\mathrm{sin}\left(\frac{1}{2}t+\frac{5\pi }{3}\right)$

amplitude: 1; period: $\text{\hspace{0.17em}}4\pi ;\text{\hspace{0.17em}}$ midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ maximum: $\text{\hspace{0.17em}}y=1\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}t=11.52;\text{\hspace{0.17em}}$ minimum: $\text{\hspace{0.17em}}y=-1\text{\hspace{0.17em}}$ occurs at $\text{\hspace{0.17em}}t=5.24;\text{\hspace{0.17em}}$ phase shift: $\text{\hspace{0.17em}}-\frac{10\pi }{3};\text{\hspace{0.17em}}$ vertical shift: 0

$f\left(x\right)=4\mathrm{sin}\left(\frac{\pi }{2}\left(x-3\right)\right)+7$

Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in [link] .

amplitude: 2; midline: $\text{\hspace{0.17em}}y=-3;\text{\hspace{0.17em}}$ period: 4; equation: $\text{\hspace{0.17em}}f\left(x\right)=2\mathrm{sin}\left(\frac{\pi }{2}x\right)-3$

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link] .

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link] .

amplitude: 2; period: 5; midline: $\text{\hspace{0.17em}}y=3;\text{\hspace{0.17em}}$ equation: $\text{\hspace{0.17em}}f\left(x\right)=-2\mathrm{cos}\left(\frac{2\pi }{5}x\right)+3$

Determine the amplitude, period, midline, and an equation involving sine for the graph shown in [link] .

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link] .

amplitude: 4; period: 2; midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ equation: $\text{\hspace{0.17em}}f\left(x\right)=-4\mathrm{cos}\left(\pi \left(x-\frac{\pi }{2}\right)\right)$

Determine the amplitude, period, midline, and an equation involving sine for the graph shown in [link] .

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link] .

amplitude: 2; period: 2; midline $\text{\hspace{0.17em}}y=1;\text{\hspace{0.17em}}$ equation: $\text{\hspace{0.17em}}f\left(x\right)=2\mathrm{cos}\left(\pi x\right)+1$

Determine the amplitude, period, midline, and an equation involving sine for the graph shown in [link] .

## Algebraic

For the following exercises, let $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\text{\hspace{0.17em}}x.$

On $\text{\hspace{0.17em}}\left[0,2\pi \right),$ solve $\text{\hspace{0.17em}}f\left(x\right)=0.$

On $\text{\hspace{0.17em}}\left[0,2\pi \right),$ solve $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{2}.$

$\frac{\pi }{6},\frac{5\pi }{6}$

Evaluate $\text{\hspace{0.17em}}f\left(\frac{\pi }{2}\right).$

On $\text{\hspace{0.17em}}\left[0,2\pi \right),f\left(x\right)=\frac{\sqrt{2}}{2}.\text{\hspace{0.17em}}$ Find all values of $\text{\hspace{0.17em}}x.$

$\frac{\pi }{4},\frac{3\pi }{4}$

On $\text{\hspace{0.17em}}\left[0,2\pi \right),$ the maximum value(s) of the function occur(s) at what x -value(s)?

On $\text{\hspace{0.17em}}\left[0,2\pi \right),$ the minimum value(s) of the function occur(s) at what x -value(s)?

$\frac{3\pi }{2}$

Show that $\text{\hspace{0.17em}}f\left(-x\right)=-f\left(x\right).\text{\hspace{0.17em}}$ This means that $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is an odd function and possesses symmetry with respect to ________________.

For the following exercises, let $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{cos}\text{\hspace{0.17em}}x.$

On $\text{\hspace{0.17em}}\left[0,2\pi \right),$ solve the equation $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{cos}\text{\hspace{0.17em}}x=0.$

$\frac{\pi }{2},\frac{3\pi }{2}$

On $\text{\hspace{0.17em}}\left[0,2\pi \right),$ solve $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{2}.$

On $\text{\hspace{0.17em}}\left[0,2\pi \right),$ find the x -intercepts of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{cos}\text{\hspace{0.17em}}x.$

$\frac{\pi }{2},\frac{3\pi }{2}$

On $\text{\hspace{0.17em}}\left[0,2\pi \right),$ find the x -values at which the function has a maximum or minimum value.

On $\text{\hspace{0.17em}}\left[0,2\pi \right),$ solve the equation $\text{\hspace{0.17em}}f\left(x\right)=\frac{\sqrt{3}}{2}.$

$\frac{\pi }{6},\frac{11\pi }{6}$

## Technology

Graph $\text{\hspace{0.17em}}h\left(x\right)=x+\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[0,2\pi \right].\text{\hspace{0.17em}}$ Explain why the graph appears as it does.

Graph $\text{\hspace{0.17em}}h\left(x\right)=x+\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[-100,100\right].\text{\hspace{0.17em}}$ Did the graph appear as predicted in the previous exercise?

The graph appears linear. The linear functions dominate the shape of the graph for large values of $\text{\hspace{0.17em}}x.$

Graph $\text{\hspace{0.17em}}f\left(x\right)=x\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[0,2\pi \right]\text{\hspace{0.17em}}$ and verbalize how the graph varies from the graph of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\text{\hspace{0.17em}}x.$

Graph $\text{\hspace{0.17em}}f\left(x\right)=x\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ on the window $\text{\hspace{0.17em}}\left[-10,10\right]\text{\hspace{0.17em}}$ and explain what the graph shows.

The graph is symmetric with respect to the y -axis and there is no amplitude because the function is not periodic.

Graph $\text{\hspace{0.17em}}f\left(x\right)=\frac{\mathrm{sin}\text{\hspace{0.17em}}x}{x}\text{\hspace{0.17em}}$ on the window $\text{\hspace{0.17em}}\left[-5\pi ,5\pi \right]\text{\hspace{0.17em}}$ and explain what the graph shows.

## Real-world applications

A Ferris wheel is 25 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. The function $\text{\hspace{0.17em}}h\left(t\right)\text{\hspace{0.17em}}$ gives a person’s height in meters above the ground t minutes after the wheel begins to turn.

1. Find the amplitude, midline, and period of $\text{\hspace{0.17em}}h\left(t\right).$
2. Find a formula for the height function $\text{\hspace{0.17em}}h\left(t\right).$
3. How high off the ground is a person after 5 minutes?
1. Amplitude: 12.5; period: 10; midline: $\text{\hspace{0.17em}}y=13.5;$
2. $h\left(t\right)=12.5\mathrm{sin}\left(\frac{\pi }{5}\left(t-2.5\right)\right)+13.5;$
3. 26 ft

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
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
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
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
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?