1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Show cos²(x) is periodic

  1. Jun 4, 2007 #1
    1. The problem statement, all variables and given/known data
    Show cos²(x) is periodic

    3. The attempt at a solution
    I don't know how :confused:, could somebody please show me step by step?
    Thanks in advance!

    edit: I've got another show/justify question.
    If f is an odd function, then show f of f is an odd function.

    Whilst I know odd of odd results in an odd function, it seems such an explanation is not suffucient.
    I have been shown this example, but can't really make any sense of it
    f o f(-x) = f(f(-x))
    = f(-f(x))
    = -f(f(x))
    = -f o f(x)
    Last edited: Jun 4, 2007
  2. jcsd
  3. Jun 5, 2007 #2

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    A function f(x) is periodic if there exists some P>0 such that f(x+P)=f(x) over the domain of f. Try rewriting cos2(x) in terms of cos(2x).
  4. Jun 5, 2007 #3


    User Avatar
    Homework Helper

    For the first one, a function f(x) is periodic, with a period of T if f(x) = f(x + T). You can use this fact to solve your first problem:

    [tex]\cos^2(x) = \cos^2(x + T)[/tex]

    A useful identity is [tex]\cos^2x = \frac{1}{2}\left(1 + \cos{2x})[/tex]. Use that identity on both sides of the above equation, then group terms in x. Since T is a constant, it must be independent of x, so something must kill all of the x terms. Find the condition on T such that this happens. If you can find such a T, then f(x) is periodic with a period of T.

    For the second one, the properties of odd functions should be enough.

    An odd function is a function such that [tex]f(-x) = -f(x)[/tex], so

    [tex]f(f(-x))=f(-f(x)) = f(-y) = -f(y) = -f(f(x))[/tex]

    Using [itex]f(x) = y[/itex] to show a little more explicitly how it works. Is that not a sufficient demonstration of what the question asks for?
  5. Jun 5, 2007 #4
    in regards to the 2nd question, I don't understand how the negative can move positions from inside the brackets to outside and then outside again.
  6. Jun 5, 2007 #5

    D H

    User Avatar
    Staff Emeritus
    Science Advisor

    You are told f(x) is an odd function: f(-x)=-f(x). You are asked to show that f(f(x)) is an odd function. Your "example" is a lot more than just an example. It is a proof of this conjecture.

    Define y=f(x). Then f(f(x)) = f(y). You want to evaluate f(f(-x)). What is f(-x)? It is -f(x)=-y, since f is an odd function. (Note: This is not true for all functions, but it is true for all odd functions by definition.) Thus f(f(-x))=f(-f(x))=f(-y). Once again using that f is odd, f(-y)=-f(y). Using y=f(x), -f(y)=-f(f(x)). Thus f(f(x))=-f(f(x)), so f(f(x)) is odd.
  7. Jun 5, 2007 #6


    User Avatar
    Science Advisor
    Homework Helper

    Why would you need an identity? It just says to show that it is periodic, not to find the smallest period. cos(x+2pi)=cos(x) -> cos^2(x+2pi)=cos^(x).
    Last edited: Jun 5, 2007
  8. Jun 5, 2007 #7
    I'm really quite confused.
    I have been shown this alternative method


    However I dont see how the cos(2pi) part dissapears as cos(2pi)=1 :confused:
  9. Jun 5, 2007 #8

    Gib Z

    User Avatar
    Homework Helper

    Umm Cos (a+b) doesnt equal cos A + cos B by the way, who ever showed you that method. But the general method was correct, show that cos x is periodic to show cos^2 x is.
  10. Jun 5, 2007 #9
    My bad, it should be

    without that step, I thought something was fishy...

    But my question still remains, how do you get from [cos(x+2pi)]² to [cos(x)]²?
  11. Jun 5, 2007 #10

    Gib Z

    User Avatar
    Homework Helper

    umm...Think of a unit circle?
  12. Jun 5, 2007 #11
    silly me....

    still struggling to comprehend how the negative can move through the brackets in the 2nd question though..
  13. Jun 5, 2007 #12

    Gib Z

    User Avatar
    Homework Helper

    Ok f(x)= - f(x) is given to us.

    We want to show f( f(x) ) = - f( f(x) )

    Ok so on the LHS replace the f(x) inside, with -f(x) since we know we can.

    Now we can f( f(x) ) = f ( -f(x) )

    Now lets replace with the f(x) with u for a second,to make things less confusing. So it becomes f (-u). f is an odd function though, so f(-u)=-f(u)

    Hence f( f(x) ) = - f( f(x) ). Done!
  14. Jun 5, 2007 #13
    I don't get this part. Why do we replace with -f(x)?
  15. Jun 5, 2007 #14

    Gib Z

    User Avatar
    Homework Helper

    f(x) = - f(x) is given to us?
  16. Jun 5, 2007 #15


    User Avatar
    Science Advisor
    Homework Helper

    Your notation...could use some help...or something.

  17. Jun 5, 2007 #16


    User Avatar
    Homework Helper

    Ok, if f is an odd function, then the following property should hold: f(-x) = -f(x), or, some also may write f(x) = -(f(-x)).

    Now, f o f(-x) = f(f(-x)), you can get this step, right?
    Since f(x) is odd, so f(-x) = -f(x). Right? So we have:
    f o f(-x) = f(f(-x)) = f(-f(x))

    Now, again, since f(x) is odd, we have: f(-f(x)) = -f(f(x)) (Notice the movement of the minus sign)

    If you still cannot see why this is true, let k = f(x), we have:
    f o f(-x) = f(f(-x)) = f(-f(x)) = f(-k) = -f(k) = -f(f(x))

    Now, the final step -f(f(x)) is actually, -f o f(x).

    So, we have shown that:
    f o f(-x) = - f o f(x), hence, f o f(x) is an odd function.

    Can you get it?

    Ok, here's some other similar problems, you can try to see if you can do it.


    Problem 1:
    f(x) is an odd function, and g(x) is an even function.
    a. Is f o g(x) odd, or even?
    b. Is g o f(x) odd, or even?

    Problem 2:
    If f o g o h(x) is an even function, and we know that f(x), and g(x) are all odd functions, what can we say about h(x)?
    Last edited: Jun 5, 2007
  18. Jun 7, 2007 #17

    Gib Z

    User Avatar
    Homework Helper

    I have a severe mental retardation, i can see that now >.< ahh i can't believe i didn't notice that...its f(-x) = -f(x) btw :(
  19. Jun 7, 2007 #18
    After many minutes thinking over it, I finally got it. Thanks to all!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook