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Thread: hyperbole trig functions

  1. #1
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    hyperbole trig functions

    i know sinh(x) = (e^x - e^-x)/2

    but what is sinhInverse(x) equal to?
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by taurus View Post
    i know sinh(x) = (e^x - e^-x)/2

    but what is sinhInverse(x) equal to?
    First of all, the terminology is "hyperbolic" not "hyperbole."

    We have a function
    $\displaystyle y = \frac{e^x - e^{-x}}{2}$

    To find the inverse we need to switch the roles of x and y:
    $\displaystyle x = \frac{e^y - e^{-y}}{2}$

    and solve for y:
    $\displaystyle 2x = e^y - e^{-y}$ <-- Multiply through by $\displaystyle e^y$

    $\displaystyle 2xe^y = e^{2y} - 1$

    $\displaystyle e^{2y} - 2xe^y - 1 = 0$

    This is a quadratic in $\displaystyle e^y$, so
    $\displaystyle e^y = \frac{2x \pm \sqrt{4x^2 + 4}}{2}$

    $\displaystyle e^y = x \pm \sqrt{x^2 + 1}$

    $\displaystyle y = ln \left ( x \pm \sqrt{x^2 + 1} \right )$

    We discard the "-" solution since the arguement of ln cannot be negative. ($\displaystyle x - \sqrt{x^2 + 1}$ is negative everywhere.)

    So finally
    $\displaystyle y = ln \left ( x + \sqrt{x^2 + 1} \right )$

    This is the inverse function to sinh(x). (The dotted line in the graph below is the line y = x.)

    -Dan
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  3. #3
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    just a question
    i got a TI 84 plus silver edition graphics calculator, does anyone know where the sinh button is if at all?
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by taurus View Post
    just a question
    i got a TI 84 plus silver edition graphics calculator, does anyone know where the sinh button is if at all?
    Try this download.

    -Dan
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