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anonymous
 4 years ago
find the period and the amplitude of the function y=6cospix
anonymous
 4 years ago
find the period and the amplitude of the function y=6cospix

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

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2As 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.

eyust707
 4 years ago
Best ResponseYou've already chosen the best response.1Yep 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
 4 years ago
Best ResponseYou've already chosen the best response.26 is the amplitude, yes. What's the period.

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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0i dont understand the period

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

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

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

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2It'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
 4 years ago
Best ResponseYou've already chosen the best response.2For 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
 4 years ago
Best ResponseYou've already chosen the best response.0ooh i think i understand now

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0lets say for y(x)=2cos4x the period would be 2pi?

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2By 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
 4 years ago
Best ResponseYou've already chosen the best response.2or 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
 4 years ago
Best ResponseYou've already chosen the best response.2For 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

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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0My teacher never taught me this and I'm trying to do the homework using the book but I find it really complicated

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2It'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
 4 years ago
Best ResponseYou've already chosen the best response.0yes but that is very simple math lol

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

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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0are you a math teacher?

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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0for y(x)=2cos4x the amplitude is 2, correct?

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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0how do i find the frequency?

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

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2I'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
 4 years ago
Best ResponseYou've already chosen the best response.0ooohh so the period of y(x)=2cos4x is pi/2? because 2pi/4 is pi/2

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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.01? because 1/2 times 1 is 2

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0i mean 1 over 1/2 is 2

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

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2Hence 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
 4 years ago
Best ResponseYou've already chosen the best response.2The lower the frequency, the higher the period. The higher the frequency, the lower the period.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0so if the period is pi/2, f=1/(pi/2)?

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

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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0I have another problem :/ E(t)=110cos(120pitpi/3)?

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

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2Yes, 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
 4 years ago
Best ResponseYou've already chosen the best response.2So 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
 4 years ago
Best ResponseYou've already chosen the best response.0ooh no no no no sorry

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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0pi/60pi=1/60=0.016666667

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

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.2Now 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.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0yes i actually got it! aah thank you soooo much!

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

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

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

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0i think its 8 but im not sure
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