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Math Help - 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
    y = \frac{e^x - e^{-x}}{2}

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

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

    2xe^y = e^{2y} - 1

    e^{2y} - 2xe^y - 1 = 0

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

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

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

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

    So finally
    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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