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The period of trigonometric functions

  1. Apr 7, 2009 #1

    ShayanJ

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    Gold Member

    Hi everyone

    Could you give me a way to calculate the period of every trigonometric functions?
    thanks
     
  2. jcsd
  3. Apr 7, 2009 #2
    Let [tex]\text{trig}\, x[/tex] be any trig function and T be it's period. The period of

    [tex]a\text{trig}\, b(x+c) + d[/tex]

    is T/b.

    The period of tan and cot is [tex]\pi[/tex] and the period of the other functions (cos, sin, sec, csc) is [tex]2\pi[/tex]. This can be remembered by the geometric definitions (sin is opp/hyp, cos is adj/hyp etc) by noticing that the ratio opp/hyp etc doesn't repeat until a complete revolution, but the ratio opp/adj and adj/opp does because (-opp)/(-adj) = opp/adj.
     
  4. Apr 7, 2009 #3
    Just to be clear, I think qntty was saying that if f(x) is a function with period T, then the period of f(ax+b) equals T/a. Is this correct?
     
  5. Apr 8, 2009 #4

    ShayanJ

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    Yes your right.
    and what about functions that are the sum of two or more trig functions and the ones that have trig functions as their nominator and/or denominator?
     
  6. Apr 8, 2009 #5
    Generally, if you have two trig functions added together, the function is no longer periodic. Take for instance:

    f(x) = sin(sqrt(2)PI*x) + cos(PI*x)
    then
    f'(x) = sqrt(2)PI*cos(sqrt(2)PI*x) - sin(PI*x)

    Periodicity implies that f(x) = f(x+T) and f'(x) = f'(x+T). However, think about it... since the two periods are incommensurate, there is no T which you can multiply by two different integers to give you multiples of the periods of each individual sin/cos. To do this would be to solve the equation

    t1 = 2PI/sqrt(2)PI = 2/sqrt(2) = sqrt(2)
    t2 = 2PI/PI=2

    T = n*t1 = m*t2

    Such as to find the smallest possible pair of numbers (n, m). But since t1 and t2 are incommensurate, and since n, m are integers, this equation has no solutions.

    In fact, functions such as sin(ax) + cos(bx) will have solutions iff the periods are commensurate, that is, they satisfy the equation I gave, and then to find the period, you find a T using the same equation.
     
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