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Math Help - Hyperbolic function identity

  1. #1
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    Hyperbolic function identity

    Express 4\cosh x+5\sinh x in the form r\sinh (x+y) giving the values of r and \tanh y.

    My problem is I don't know which identities to use to change forms, I've recently started on this hyperbolic functions and am not very familiar with them yet.
    Thanks!
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  2. #2
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    Hello, arze!

    We need:

    . . \sinh(A + B) \:=\:\sinh A\cosh B + \cosh A\sinh B

    . . \tanh x \:=\:\dfrac{\sinh x}{\cosh x}



    \text{Express }4\cosh x+5\sinh x\text{ in the form }r\sinh (x+y)

    \text{Give the values of }r\text{ and }\tanh y

    Let: . Z \;=\;4\cosh x + 5\sinh x

    Divide by 3: . \dfrac{Z}{3} \;=\;\frac{4}{3}\cosh x + \frac{5}{3}\sinh x .[1]


    Let: . \sinh y = \frac{4}{3},\;\cosh y = \frac{5}{3}


    Substitute into [1]: . \dfrac{Z}{3} \;=\;\cosh y \sinh x + \cosh y \sinh x

    . . . . And we have: . \dfrac{Z}{3} \;=\;\sinh(x + y)


    Hence: . Z \;=\;3\sinh(x + y)\;\text{ where }y \:=\:\sinh^{-1}\left(\frac{4}{3}\right)


    Therefore: . r \:=\:3

    . . . and: . \tanh y \:=\:\dfrac{\sinh y}{\cosh y} \:=\:\dfrac{\frac{4}{3}}{\frac{5}{3}} \:=\:\dfrac{4}{5}
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