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Thread: Hyperbolic function questions

  1. #1
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    Hyperbolic function questions

    Hi

    I need help on the following questions:

    1) Express sinh(x+y) in terms of sinh(x), sinh(y), cosh(x), cosh(y)


    2) Given cosh $\displaystyle u=\frac{5}{4}$ find values of sinh(u)

    This is what i have done:

    made $\displaystyle u=\frac{5}{4}$

    so sinh(u) = 1.89.

    What is wrong??

    P.S
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  2. #2
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    Quote Originally Posted by Paymemoney View Post
    Hi

    I need help on the following questions:

    1) Express sinh(x+y) in terms of sinh(x), sinh(y), cosh(x), cosh(y)


    2) Given cosh $\displaystyle u=\frac{5}{4}$ find values of sinh(u)
    You need to know the following:

    $\displaystyle sinh(x) = \frac{e^x - e^{-x}}{2}$ $\displaystyle cosh(x) = \frac{e^x + e^{-x}}{2}$

    $\displaystyle e^x = \cosh{x} + \sinh{x}$

    $\displaystyle \cosh{(-x)} = \cosh{x}$

    $\displaystyle \sinh{(-x)} = -\sinh{x}$
    ------------------------------------------------------
    1)$\displaystyle sinh(x+y) = \frac{e^{x+y} - e^{-(x+y)}}{2} = \frac{e^{x}e^{y} - e^{-x}e^{-y}}{2} =$ $\displaystyle 1/2[(\cosh{x}+\sinh{x})(\cosh{(y)}+\sinh{(y)})$ $\displaystyle - (\cosh{(-x)}+\sinh{(-x)})(\cosh{(-y)}+\sinh{(-y)})]=$ $\displaystyle \sinh{x} \cosh {y} + \cosh{x} \sinh{y}$

    2) Is the question: Given,

    $\displaystyle cosh(u) = \frac{e^{u} + e^{-u}}{2} =5/4?$

    Or is it $\displaystyle u=5/4?$
    Last edited by Anonymous1; Apr 4th 2010 at 07:59 PM.
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  3. #3
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    Quote Originally Posted by Anonymous1 View Post
    2) Is the question: Given,

    $\displaystyle cosh(u) = \frac{e^{u} + e^{-u}}{2} =5/4?$
    its given like that ^^
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  4. #4
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    Quote Originally Posted by Paymemoney View Post
    its given like that ^^
    In that case sub this identity:

    $\displaystyle e^x = \cosh{x} + \sinh{x}$

    into the definition of cosh, set it equal to 5/4, and put the whole thing in terms of sinh.
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  5. #5
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    Quote Originally Posted by Paymemoney View Post
    [snip]

    2) Given cosh $\displaystyle u=\frac{5}{4}$ find values of sinh(u)

    This is what i have done:

    made $\displaystyle u=\frac{5}{4}$

    so sinh(u) = 1.89.

    What is wrong??

    P.S
    Several approaches are possible. The simplest is to use the identity $\displaystyle \cosh^2 u - \sinh^2 u = 1$.
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  6. #6
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    this is what i have done, someone tell me if this is a correct way of doing it

    $\displaystyle \frac{(cosh(u)+sinh(u))+(cosh(-u)+sinh(-u))}{2} = \frac{5}{4}$

    $\displaystyle 10 = 4(cosh(u)+sinh(u))+(cosh(-u)+sinh(-u))$

    $\displaystyle 2.5 = 2cosh(u)$

    $\displaystyle 1.25 = \frac{1}{sinh(u)}$

    $\displaystyle sinh(u) = \frac{1}{1/1.25} = 0.8$ my answer is slighlty difference to the book's answers of $\displaystyle \pm0.75$.
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