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Is f(x) = cos^2(x) + sin^2(x) a periodic function?

  1. Apr 28, 2013 #1
    1. The problem statement, all variables and given/known data
    Is f(x) = cos^2(x) + sin^2(x) a periodic function?


    2. Relevant equations
    sin^2(x) + cos^2(x) = 1


    3. The attempt at a solution
    This question is just something that randomly came to my mind (not a hw problem). I know cos^2(x) and sin^2(x) are both periodic functions, but is sin^2(x) + cos^2(x) a periodic function too? If so, what would be its fundamental frequency?
     
  2. jcsd
  3. Apr 28, 2013 #2

    Zondrina

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    Homework Helper

    No, sin^2(x) + cos^2(x) = 1 which we know is not periodic, but constant.
     
  4. Apr 28, 2013 #3
    Indeed, as Zondrina mentioned, it is not periodic. Typically, we say a function f(x) is periodic if there is a smallest positive integer P≠0 for which f(x+P)=f(x). The number P is then the period. Since sin^2(x)+cos^2(x)=1:=1(x), it is not periodic, because 1(x+P)=1=1(x) for any P (so in particular, there isn't a smallest one).
     
    Last edited: Apr 28, 2013
  5. Apr 28, 2013 #4
    Technically a constant function is periodic but has no fundamental period, but it is kind of silly to refer to it in such a way.
     
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