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Math Help - Period of sums/multiplied sines/cosines

  1. #1
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    Cool Period of sums/multiplied sines/cosines

    Hello

    I am trying to revise some rules regarding the period of sines (or cosines) when (a) they are added , and (b) they are multiplied. I haven't had too much luck Googling this topic.

    For example, what would be the fundamental period of:

    y=cos(2.pi.2x) + cos(2.pi.3x)

    Or, if we multiply:

    y=cos(2.pi.2x).cos(2.pi.3x)

    People have suggested to me that I use trig. identities (or complex exponentials) but I am sure there is a more intuitive way such as considering the phases between both parts.

    Any ideas appreciated !

    Cheers

    Gabber
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  2. #2
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    Re: Period of sums/multiplied sines/cosines

    Quote Originally Posted by gabber808 View Post
    People have suggested to me that I use trig. identities (or complex exponentials) but I am sure there is a more intuitive way such as considering the phases between both parts.
    Consider f(x) = cos(2\pi .2x) + cos(2\pi .3x) if f(x) is periodic then f(x+T) = f(x) where T is a period.

    Now we have cos(2\pi .2(x+T)) + cos(2\pi .3(x+T)) = cos(2\pi .2x) + cos(2\pi .3x) as cosine is a function with periodicity 2\pi we must have 2\pi(2T) and 2\pi(3T) are both of the form 2m\pi where m is an integer not necessarily same in both the cases. In first case its enough if T = \frac{m}{2} ,and in the second case if T = \frac{n}{3} where m,n \in Z where Z is the set of integers. Therefore if T=1 to be the fundamental period then we are fine with it.

    Multiplication of functions can be handled on the same lines.

    Now try solving the following problem instead
    f(x) = cos(2\pi .\frac{x}{2}) + cos(2\pi .\frac{x}{3})

    Kalyan.
    Last edited by kalyanram; July 10th 2011 at 06:47 AM.
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    Thumbs up Re: Period of sums/multiplied sines/cosines

    Thank you Kaylan

    So it looks like the period will be the LCM (lowest common multiple) of the individual periods?

    I think the answer to your question is SIX

    Do you have any good ideas how to find the period when the cosines are multiplied?

    Thanks again

    Gabber
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  4. #4
    Member kalyanram's Avatar
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    Re: Period of sums/multiplied sines/cosines

    So it looks like the period will be the LCM (lowest common multiple) of the individual periods?

    I think the answer to your question is SIX
    That's true period in the case of sums is the LCM of individual periods.

    Do you have any good ideas how to find the period when the cosines are multiplied?
    Now in the case when cosines are multiplied or for that matter any powers of sines or cosines can be handled by transforming them to sums of sines and cosines.

    Kalyan.
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  5. #5
    Member kalyanram's Avatar
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    Re: Period of sums/multiplied sines/cosines

    Here are few examples
    f(x) = 2cos(2\pi . \frac{x}{2}).cos(2\pi . \frac{x}{3}) \Rightarrow f(x) = cos(2\pi . \frac{5x}{6}) + cos(2\pi . \frac{x}{6}) as you can see since the LCM has already been obtained during the sum and difference of angles the approach does not change drastically. However in the following example the approach varies

    f(x) = cos^3 (2\pi . \frac{x}{12}) \Rightarrow f(x) = \frac{1}{4} (3cos(2\pi x) - cos(2\pi . \frac{x}{4})) , T = 4.

    Kalyan
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    Re: Period of sums/multiplied sines/cosines

    your question is:
    --------------------------
    For example, what would be the fundamental period of:
    y=cos(2.pi.2x) + cos(2.pi.3x)
    Or, if we multiply:
    y=cos(2.pi.2x).cos(2.pi.3x)
    -----------------------------
    my anwer is:
    in both cases the total period is
    x=(0;1)
    that is the least common multiple of the two periods.
    in fact the 1st period is x=(0;1/2)
    and the 2nd period is x=(0;1/3)
    therefore the range x=(0;1) is the least common multiple of the two single ranges.
    bye-bye
    mike999
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    Re: Period of sums/multiplied sines/cosines

    Thanks everybody

    Mike, you seem to be saying that this LCM method is also true for procucts of sines (or cosines), not just when they are added. So is this generally true?

    Kalyan, in your cos-cubed example, after you transform it, we have periods of 1 and 4. So again it is the LCM.

    Thanks again

    Matt
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    Re: Period of sums/multiplied sines/cosines

    Hi Matt,

    Mike, you seem to be saying that this LCM method is also true for procucts of sines (or cosines), not just when they are added. So is this generally true?

    Kalyan, in your cos-cubed example, after you transform it, we have periods of 1 and 4. So again it is the LCM.
    The idea of taking an LCM is valid for the case of sum or multiplication of functions. All I was suggestting is that you have to consider taking the transformations when multiplication of functions is involved as in the case of cos^2 \theta, cos^3 \theta are involved.

    Kalyan.
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    Re: Period of sums/multiplied sines/cosines

    Hello again

    Thank you for all your help so far Kalyan

    I am glad you agree about the LCM idea !

    However, I have just been trying an online graphing program.

    Firstly I tried cos(2pi*(x/2))*cos(2pi*(x/3)). As expected I can see that the pattern repeats every 6 units.

    Then I tried cos(2pi*x)*cos(2pi*(x/3)). Maybe I am wrong, but it looks like a period of 1.5. There is ALSO a period of 3. But the lowest common multiple (LCM) idea does not suggest that !

    Does this mean that perhaps the LCM idea is not the way to go and really we should always use trig identies to get the period when our sines or cosines are multiplied?
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  10. #10
    Member kalyanram's Avatar
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    Re: Period of sums/multiplied sines/cosines

    Hi Gbber808,
    Then I tried cos(2pi*x)*cos(2pi*(x/3)). Maybe I am wrong, but it looks like a period of 1.5. There is ALSO a period of 3. But the lowest common multiple (LCM) idea does not suggest that !
    Well all our previous examples we dealt with simple cases as of the type \frac{x}{m} , \frac{x}{n} where we took the lcm(m,n) in the case f(x) = cos(2\pi x).cos(2 \pi \frac{x}{3}) = \frac{1}{2}(cos(2\pi \frac{4x}{3}) + cos(2 \pi \frac{2x}{3})) and we see that its enough for T =n \frac{3}{2} for cos(2\pi \frac{4x}{3}) , cos(2\pi \frac{2x}{3}) to be periodic.

    Kalyan.
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  11. #11
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    Re: Period of sums/multiplied sines/cosines

    Another way to look at it is that you have taken the lcm(1,3) i.e LCM of denominators and divided it by HCF of numerators hcf(2,4)
    T = 3/2.

    Kalyan.
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    Re: Period of sums/multiplied sines/cosines

    Sorry for the late reply. Apart from lots of work I have also been revising my trig identities. It was a long ago I learnt those. Thanks again for all of your help Kalyanram.
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