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Math Help - Hyperbolic Function Proof

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    Hyperbolic Function Proof

    Prove that \mathrm{sinh}(i\pi-\theta)=\mathrm{sinh}\theta.

    Thanks in advance.
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  2. #2
    Moo
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    Hi !

    Quote Originally Posted by Air View Post
    Prove that \mathrm{sinh}(i\pi-\theta)=\mathrm{sinh}\theta.

    Thanks in advance.
    \sinh x=\frac{e^x-e^{-x}}{2}

    Then, replace x by i \pi-\theta, and simplify, using this formula : e^{i \pi}=\cos \pi+i \sin \pi=-1

    -------------

    or :

    \sinh(x)=i \sin(ix)

    ---> \sinh(i \pi-\theta)=i \sin(-\pi-i \theta)=i \sin(-\pi+2 \pi-i \theta)=i \sin(\pi-i\theta)

    But \sin(\pi-x)=\sin (x)

    So i \sin(\pi-i \theta)=\dots


    Last edited by Moo; June 18th 2008 at 01:53 PM. Reason: edited
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    Quote Originally Posted by Moo View Post
    e^{i \pi}=\cos \pi+i \sin \pi=-1
    How did you know that the formula equals -1 or is it just a standard answer?
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    Moo
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    Quote Originally Posted by Air View Post
    How did you know that the formula equals -1 or is it just a standard answer?
    That's because \cos \pi=-1 and \sin \pi=0

    You should know that e^{ix}=\cos x+i \sin x


    Or just know the formula : e^{i\pi}+1=0 which can be found in several signatures in this forum (Chris L T521, kalagota, ... ?)
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    So... I did:

    \frac{e^{i\pi-\theta} - e^{-(i\pi-\theta)}}{2}

    I can simplify e^{i\pi - \theta}=-1e^{\theta} but what about e^{-(i\pi-\theta)}?
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  6. #6
    Moo
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    Quote Originally Posted by Air View Post
    So... I did:

    \frac{e^{i\pi-\theta} - e^{-(i\pi-\theta)}}{2}

    I can simplify e^{i\pi - \theta}=-1e^{{\color{red}-}\theta} but what about e^{-(i\pi-\theta)}?
    hehe, that's the way (little mistake in red)

    e^{-(i \pi-\theta)}=e^{-i\pi+\theta}=e^{-i\pi} \cdot e^{\theta}=\frac{1}{e^{i \pi}} \cdot e^{\theta}

    << I used the rule : a^{b+c}=a^b \cdot a^c
    And a^{-b}=\frac{1}{a^b}

    be careful ^^
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  7. #7
    Super Member flyingsquirrel's Avatar
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    Hello
    Quote Originally Posted by Moo View Post
    [...]
    {\color{red}i}\sinh(x)=\sin(ix)
    [...]
    "Little mistake in red."

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  8. #8
    Moo
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    Quote Originally Posted by flyingsquirrel View Post
    Hello

    "Little mistake in red."

    I forgot that
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