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eli123
Group Title
find the period and the amplitude of the function y=6cospix
 2 years ago
 2 years ago
eli123 Group Title
find the period and the amplitude of the function y=6cospix
 2 years ago
 2 years ago

This Question is Closed

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
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?
 2 years ago

eyust707 Group TitleBest ResponseYou've already chosen the best response.1
in other words: cos(0) = 1 cos(2pi) = 1 cos(4pi) = 1 cos(6pi) = 1 see a pattern?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
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.
 2 years ago

eyust707 Group TitleBest ResponseYou've already chosen the best response.1
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
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
6 is the amplitude, yes. What's the period.
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
@eyust707, you've got this one. Thanks.
 2 years ago

eyust707 Group TitleBest ResponseYou've already chosen the best response.1
any time James
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
i dont understand the period
 2 years ago

imranmeah91 Group TitleBest ResponseYou've already chosen the best response.0
cos(a x) any thing in front of x , in this case divide period= 2pi/a so cos(2 x) period = 2pi/2 = pi
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
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)?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
As eyust noted above, cos(0) = 1 cos(2pi) = 1 cos(4pi) = 1 cos(6pi) = 1 So what is T?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
i.e., what is the period for the function g(x) = cos(x)?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
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) ?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
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?
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
ooh i think i understand now
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
lets say for y(x)=2cos4x the period would be 2pi?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
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.
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
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.
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
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
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
OMG this is hard!
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
No, it's just new for you. Do it a few more times and it'll be easy for you.
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
My teacher never taught me this and I'm trying to do the homework using the book but I find it really complicated
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
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.
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
yes but that is very simple math lol
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
For me, the questions you're asking are also very simple.
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
but there was a time when they were new for me too. Just stick with it, and do a few more problems!
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
are you a math teacher?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
Former University Lecturer
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
for y(x)=2cos4x the amplitude is 2, correct?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
Yes, the amplitude of y(x) = 2cos(4x) is 2, correct.
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
how do i find the frequency?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
What's the definition of frequency, f?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
the rate of what?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
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?
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
ooohh so the period of y(x)=2cos4x is pi/2? because 2pi/4 is pi/2
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
So now, returning to my last post and your question on frequency. What is the relationship between period T and frequency f?
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
1? because 1/2 times 1 is 2
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
i mean 1 over 1/2 is 2
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
Yes, so f = 1/T. Or T = 1/f
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
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.
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
Yes T = period.
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
The lower the frequency, the higher the period. The higher the frequency, the lower the period.
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
so if the period is pi/2, f=1/(pi/2)?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
Yes. If T = pi/2, then f = 2/pi.
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
Which means every unit of time there are 2/pi complete oscillations.
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
I have another problem :/ E(t)=110cos(120pitpi/3)?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
So you should know enough now to try and figure this out. First, what's the amplitude?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
Yes, exactly. Now remember that the period of cos and sin is \( 2\pi \). I.e., \[ \cos(x + 2\pi) = \cos(x) \ \ \ \ \hbox{ and } \ \ \ \sin(x + 2\pi) = \sin(x) \]
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
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?
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
ooh no no no no sorry
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
If 120πT=2π, then T = ...
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
pi/60pi=1/60=0.016666667
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
T = 1/60. Don't write the decimal expansion unless you really, really have to. Always messy.
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
Now that looked complicated, but you can always just read it off from the coefficient of t. Given E(t)=110cos(120pi.tpi/3), the coefficient of t is 120pi. Hence the period is T = 2pi/120pi = 1/60.
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
yes i actually got it! aah thank you soooo much!
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
What is the frequency of E(t). What is the value of f for that function?
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
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.
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
If the units of t are seconds, E(t) has a complete oscillation every 1/60 seconds.
 2 years ago

JamesJ Group TitleBest ResponseYou've already chosen the best response.2
Try and answer the first part of this problem: http://openstudy.com/#/updates/4ed01d1de4b04e045af4e631
 2 years ago

eli123 Group TitleBest ResponseYou've already chosen the best response.1
i think its 8 but im not sure
 2 years ago
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