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Math Help - Explain why the equation cos x = x has at least one solution?

  1. #1
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    Red face Explain why the equation cos x = x has at least one solution?

    Hello again everyone

    Does anyone know why the equation cos x = x has at least one solution?

    Thanks!
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  2. #2
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    Re: Explain why the equation cos x = x has at least one solution?

    There are several ways to look at this question.

    First, at x=0, cos(x) is equal to 1, and x is equal to just 0 (since x=0=0). So at this point cos(x) > x. But the cosine function oscillates forever while the graph of y=x rises continuously to infinity. Therefore, at some point, the graph of y=x will have to "catch up" to the graph of y=cos(x).

    You can look at it geometrically as well. If you draw a unit circle, when cos(x)=x, the arc length of the angle x will be equal to the x-coordinate.

    If you want a more rigorous answer, if we define f(x) = cos(x) - x, this function takes on both positive and negative values. (f(0) = 1, f(2*Pi) = (1-2*Pi). Due to the intermediate value theorem, f(x) must somewhere take on the value 0, which means that cos(x) will equal x, since their difference is 0.
    Last edited by SworD; September 2nd 2012 at 11:34 PM.
    Thanks from Nora314
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    Re: Explain why the equation cos x = x has at least one solution?

    That was a beautiful answer, thank you
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