find the period and the amplitude of the function y=6cospix

- anonymous

find the period and the amplitude of the function y=6cospix

- chestercat

See more answers at brainly.com

At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

Get this expert

answer on brainly

SEE EXPERT ANSWER

Get your **free** account and access **expert** answers to this

and **thousands** of other questions

- JamesJ

You can figure out the period from one time cos(pi.x) = 1 to the next time.
So cos(pi.x) = 1 when x = 0, as cos(pi.0) = cos(0) = 1. What is the next value of x for which
cos(pi.x) = 1?

- eyust707

in other words:
cos(0) = 1
cos(2pi) = 1
cos(4pi) = 1
cos(6pi) = 1
see a pattern?

- JamesJ

As for the amplitude, that is the maximum value of this function. y = f(x) = 6 cos(pi.x). What is it's maximum value?
Hint: it occurs when cos(pi.x) has its maximum value.

Looking for something else?

Not the answer you are looking for? Search for more explanations.

## More answers

- anonymous

6?

- eyust707

Yep so basically like james said, the amplitude is = to the maxium valuse that we can make 6cos(pi*x)
Since the 6 is a constant the only thing we can change is the x. We need to change the x to something that will make "cos(pi*x)" as big as possible.
if we plug in all the possible values around the unit circle you will find that cos never gets bigger than 1

- JamesJ

6 is the amplitude, yes.
What's the period.

- JamesJ

@eyust707, you've got this one. Thanks.

- eyust707

any time James

- anonymous

i dont understand the period

- anonymous

cos(a x)
any thing in front of x , in this case
divide
period= 2pi/a
so cos(2 x)
period = 2pi/2 = pi

- JamesJ

The period of a function f is the smallest number T for which
f(x + T) = f(x).
For the function f(x) = 6 cos(pi.x), it is therefore the smallest number T such that
6cos(pi(x+T)) = 6cos(pi.x)
that is
cos(pi.(x+T)) = cos(pi.x)
Now if the pi wasn't there, draw the function g(x) = cos(x). For what value of T is it the case that g(x + T) = g(x)? i.e.,
cos(x + T) = cos(x)?

- JamesJ

As eyust noted above,
cos(0) = 1
cos(2pi) = 1
cos(4pi) = 1
cos(6pi) = 1
So what is T?

- JamesJ

i.e., what is the period for the function g(x) = cos(x)?

- JamesJ

It's clear that the period of g(x) = cos(x) is T = 2pi.
Now that being the case, what is the period of the function f(x) where
f(x) = 6 cos(pi.x) ?
I.e., for what value of T is it the case that
f(x+T) = f(x)
cos(pi(x+T)) = cos(pi.x) ?

- JamesJ

For example, for x = 0,
cos(pi(0+T) = cos(pi.0)
i.e.,
cos(piT) = cos(0) = 1
i.e.,
cos(pi.T) = 1.
What is the smallest such number T so that is the case?

- anonymous

ooh i think i understand now

- anonymous

lets say for y(x)=-2cos4x the period would be 2pi?

- JamesJ

No.

- JamesJ

By definition, it is the smallest number T such that
y(x+T) = y(x)
i.e.,
-2 cos(4(x+T)) = - 2 cos(4x)
i.e.,
cos(4x + 4T) = cos(4x)
Now cos has period 2pi. Hence
4T = 2pi
or T = pi/2.
Therefore the period of y(x) is T = pi/2.

- JamesJ

or in other words, as imran was just writing, if you have a function
f1(x) = sin(ax) or f2(x) = cos(ax),
as both sin and cos have period 2pi, it follows that the period of both f1 and f2 is
2pi/a.

- JamesJ

For example, the period T of f1 is the smallest number T such that
f1(x + T) = f1(x)
i.e.,
sin(a(x+T)) = sin(ax)
i.e.,
sin(ax + aT) = sin(ax)
i.e., aT = 2pi, because the period of sin is 2pi
i.e., T = 2pi/a

- anonymous

OMG this is hard!

- JamesJ

No, it's just new for you. Do it a few more times and it'll be easy for you.

- anonymous

My teacher never taught me this and I'm trying to do the homework using the book but I find it really complicated

- JamesJ

It's like the first time you saw algebra. It seemed hard, but now you can solve equations like
2x + 4 = 6
In your sleep.

- anonymous

yes but that is very simple math lol

- JamesJ

For me, the questions you're asking are also very simple.

- JamesJ

but there was a time when they were new for me too. Just stick with it, and do a few more problems!

- anonymous

are you a math teacher?

- JamesJ

Former University Lecturer

- anonymous

awesome

- anonymous

for y(x)=-2cos4x the amplitude is 2, correct?

- JamesJ

Yes, the amplitude of y(x) = -2cos(4x) is 2, correct.

- anonymous

how do i find the frequency?

- JamesJ

What's the definition of frequency, f?

- anonymous

the rate?

- JamesJ

the rate of what?

- anonymous

of the wave

- JamesJ

I'll tell you. If a function is periodic, i.e., oscillates, it has a period, T such that
f(t+T) = f(t)
The frequency is the number of complete oscillations per unit of time.
For example if T = 1, then there would be one oscillation per unit of time, seconds say. I.e., f = 1.
If T = 2, there would be 1/2 an oscillation per second. I.e., f = 1/2.
If T = 1/2, there would 2 oscillations per second, f = 2.
Given all that, what is the relation between T and f?

- anonymous

ooohh so the period of y(x)=-2cos4x is pi/2? because 2pi/4 is pi/2

- JamesJ

yes.

- anonymous

omg yay lol

- JamesJ

So now, returning to my last post and your question on frequency. What is the relationship between period T and frequency f?

- anonymous

1? because 1/2 times 1 is 2

- anonymous

i mean 1 over 1/2 is 2

- JamesJ

Yes, so f = 1/T. Or T = 1/f

- anonymous

T =period?

- JamesJ

Hence the higher the period, the lower the frequency. Makes sense because if the period is longer, there can be less complete oscillations in a second.
Or the lower the period, the higher the frequency.

- JamesJ

Yes T = period.

- JamesJ

The lower the frequency, the higher the period.
The higher the frequency, the lower the period.

- anonymous

so if the period is pi/2, f=1/(pi/2)?

- JamesJ

Yes. If T = pi/2, then f = 2/pi.

- JamesJ

Which means every unit of time there are 2/pi complete oscillations.

- anonymous

got it :D

- anonymous

I have another problem :/ E(t)=110cos(120pit-pi/3)?

- JamesJ

So you should know enough now to try and figure this out. First, what's the amplitude?

- anonymous

110

- JamesJ

Yes, exactly.
Now remember that the period of cos and sin is \( 2\pi \). I.e.,
\[ \cos(x + 2\pi) = \cos(x) \ \ \ \ \hbox{ and } \ \ \ \sin(x + 2\pi) = \sin(x) \]

- JamesJ

So go back to first principles to find the period T of your new function E(t).
It is the number T such that
E(t+T) = E(t)
i.e.,
\[ 110 \cos(120\pi(t+T)-\pi/3) = 110 \cos(120\pi t-\pi/3) \]
i.e.,
\[ \cos(120\pi(t+T)-\pi/3) = \cos(120\pi t - \pi/3) \]
i.e.,
\[ \cos(120\pi t - \pi/3 + 120\pi T) = \cos(120\pi t - \pi/3) \]
i.e.,
\[ 120\pi T = 2\pi\]
because the period of cos is \( 2\pi \)
Hence T = ... what?

- anonymous

60?

- JamesJ

Noo...

- anonymous

ooh no no no no sorry

- JamesJ

If 120πT=2π, then T = ...

- anonymous

2pi/120pi

- JamesJ

Simplify

- anonymous

pi/60pi=1/60=0.016666667

- JamesJ

T = 1/60. Don't write the decimal expansion unless you really, really have to. Always messy.

- JamesJ

Now that looked complicated, but you can always just read it off from the coefficient of t.
Given
E(t)=110cos(120pi.t-pi/3),
the coefficient of t is 120pi.
Hence the period is T = 2pi/120pi = 1/60.

- anonymous

yes i actually got it! aah thank you soooo much!

- JamesJ

What is the frequency of E(t). What is the value of f for that function?

- anonymous

60

- JamesJ

Yes, exactly. If the units of t are seconds, E(t) has 60 complete oscillations per second.
If the units of t are days, then E(t) has 60 complete oscillations per day. Etc.

- JamesJ

If the units of t are seconds, E(t) has a complete oscillation every 1/60 seconds.

- anonymous

alright

- JamesJ

Try and answer the first part of this problem:
http://openstudy.com/#/updates/4ed01d1de4b04e045af4e631

- anonymous

i think its 8 but im not sure

Looking for something else?

Not the answer you are looking for? Search for more explanations.