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Math Help - x = tan(x)

  1. #1
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    x = tan(x)

    Can some explain to me why xcos(x) - sin(x) will = 0 when x = tan(x)?

    Thanks.
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    Re: x = tan(x)

    Quote Originally Posted by alyosha2 View Post
    Can some explain to me why xcos(x) - sin(x) will = 0 when x = tan(x)?
    \tan(x)=\frac{\sin(x)}{\cos(x)}
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    Re: x = tan(x)

    I understand that, but I don't see how the answer follows from this.
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    Re: x = tan(x)

    I just don't see it. why is the output of the function needed to be the same as the input? How does that help? What effect does cos or sin have on sinx/cosx? I can't seem to find a pattern. What's the releveance of having tanx * cos(tanx)? Am I thinking about it in the right way? If x and tanx are then interchangeable, what is it that is so special about the value of x at the points where x = tanx that makes it work? I've spend two hours trying to figure this out and I am just completely lost.
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    Re: x = tan(x)

    I've had an idea. Is it because x and tanx are interchangeable you can use tanx only as a replacement for the first x leaving you with sinx/cosx * cosx - sinx = sinx - sinx = 0?
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    Re: x = tan(x)

    Quote Originally Posted by alyosha2 View Post
    Can some explain to me why xcos(x) - sin(x) will = 0 when x = tan(x)?

    Thanks.
    \displaystyle \begin{align*} x &= \tan{x} \\ x &= \frac{\sin{x}}{\cos{x}} \\ x\cos{x} &= \sin{x} \\ x\cos{x} - \sin{x} &= 0 \end{align*}
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    Re: x = tan(x)

    Quote Originally Posted by Prove It View Post
    \displaystyle \begin{align*} x &= \tan{x} \\ x &= \frac{\sin{x}}{\cos{x}} \\ x\cos{x} &= \sin{x} \\ x\cos{x} - \sin{x} &= 0 \end{align*}
    Almost. Note that the relation is NOT true when x = \pi /2.

    @alyosha2 Why does this problem arise with Prove It's proof?

    -Dan
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    Re: x = tan(x)

    Quote Originally Posted by topsquark View Post
    Almost. Note that the relation is NOT true when x = \pi /2.
    This was all assuming that x= tan(x) which is NOT true when x= \pi/2.

    @alyosha2 Why does this problem arise with Prove It's proof?

    -Dan
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    Re: x = tan(x)

    Quote Originally Posted by topsquark View Post
    Almost. Note that the relation is NOT true when x = \pi /2.

    @alyosha2 Why does this problem arise with Prove It's proof?

    -Dan
    I had assumed we were working in the maximal domain implied by the problem: \displaystyle \begin{align*} x \in \mathbf{R} \backslash \frac{n\pi}{2}, n \in \mathbf{Z} \end{align*}.
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    Re: x = tan(x)

    @ Halls and Prove It, good point.

    -Dan
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