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

1. JamesJ Group Title

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. eyust707 Group Title

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

3. JamesJ Group Title

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.

4. eli123 Group Title

6?

5. eyust707 Group Title

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

6. JamesJ Group Title

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

7. JamesJ Group Title

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

8. eyust707 Group Title

any time James

9. eli123 Group Title

i dont understand the period

10. imranmeah91 Group Title

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

11. JamesJ Group Title

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)?

12. JamesJ Group Title

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

13. JamesJ Group Title

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

14. JamesJ Group Title

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) ?

15. JamesJ Group Title

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?

16. eli123 Group Title

ooh i think i understand now

17. eli123 Group Title

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

18. JamesJ Group Title

No.

19. JamesJ Group Title

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.

20. JamesJ Group Title

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.

21. JamesJ Group Title

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

22. eli123 Group Title

OMG this is hard!

23. JamesJ Group Title

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

24. eli123 Group Title

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

25. JamesJ Group Title

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.

26. eli123 Group Title

yes but that is very simple math lol

27. JamesJ Group Title

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

28. JamesJ Group Title

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

29. eli123 Group Title

are you a math teacher?

30. JamesJ Group Title

Former University Lecturer

31. eli123 Group Title

awesome

32. eli123 Group Title

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

33. JamesJ Group Title

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

34. eli123 Group Title

how do i find the frequency?

35. JamesJ Group Title

What's the definition of frequency, f?

36. eli123 Group Title

the rate?

37. JamesJ Group Title

the rate of what?

38. eli123 Group Title

of the wave

39. JamesJ Group Title

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?

40. eli123 Group Title

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

41. JamesJ Group Title

yes.

42. eli123 Group Title

omg yay lol

43. JamesJ Group Title

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

44. eli123 Group Title

1? because 1/2 times 1 is 2

45. eli123 Group Title

i mean 1 over 1/2 is 2

46. JamesJ Group Title

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

47. eli123 Group Title

T =period?

48. JamesJ Group Title

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.

49. JamesJ Group Title

Yes T = period.

50. JamesJ Group Title

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

51. eli123 Group Title

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

52. JamesJ Group Title

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

53. JamesJ Group Title

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

54. eli123 Group Title

got it :D

55. eli123 Group Title

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

56. JamesJ Group Title

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

57. eli123 Group Title

110

58. JamesJ Group Title

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

59. JamesJ Group Title

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?

60. eli123 Group Title

60?

61. JamesJ Group Title

Noo...

62. eli123 Group Title

ooh no no no no sorry

63. JamesJ Group Title

If 120πT=2π, then T = ...

64. eli123 Group Title

2pi/120pi

65. JamesJ Group Title

Simplify

66. eli123 Group Title

pi/60pi=1/60=0.016666667

67. JamesJ Group Title

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

68. JamesJ Group Title

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.

69. eli123 Group Title

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

70. JamesJ Group Title

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

71. eli123 Group Title

60

72. JamesJ Group Title

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.

73. JamesJ Group Title

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

74. eli123 Group Title

alright

75. JamesJ Group Title