Results 1 to 8 of 8

Thread: Hyperbolic Function Proof

  1. #1
    Super Member
    Joined
    Sep 2007
    Posts
    528
    Awards
    1

    Hyperbolic Function Proof

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

    Thanks in advance.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hi !

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

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

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

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

    or :

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

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

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

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


    Last edited by Moo; Jun 18th 2008 at 12:53 PM. Reason: edited
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Sep 2007
    Posts
    528
    Awards
    1
    Quote Originally Posted by Moo View Post
    $\displaystyle e^{i \pi}=\cos \pi+i \sin \pi=-1$
    How did you know that the formula equals -1 or is it just a standard answer?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    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 $\displaystyle \cos \pi=-1$ and $\displaystyle \sin \pi=0$

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


    Or just know the formula : $\displaystyle e^{i\pi}+1=0$ which can be found in several signatures in this forum (Chris L T521, kalagota, ... ?)
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Sep 2007
    Posts
    528
    Awards
    1
    So... I did:

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

    I can simplify $\displaystyle e^{i\pi - \theta}=-1e^{\theta}$ but what about $\displaystyle e^{-(i\pi-\theta)}$?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Quote Originally Posted by Air View Post
    So... I did:

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

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

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

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

    be careful ^^
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member flyingsquirrel's Avatar
    Joined
    Apr 2008
    Posts
    802
    Hello
    Quote Originally Posted by Moo View Post
    [...]
    $\displaystyle {\color{red}i}\sinh(x)=\sin(ix)$
    [...]
    "Little mistake in red."

    Follow Math Help Forum on Facebook and Google+

  8. #8
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Quote Originally Posted by flyingsquirrel View Post
    Hello

    "Little mistake in red."

    I forgot that
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Hyperbolic proof
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: Sep 24th 2011, 02:40 PM
  2. Replies: 2
    Last Post: Mar 20th 2011, 01:29 AM
  3. Hyperbolic Function proof
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Feb 28th 2010, 12:33 AM
  4. Hyperbolic Function Derivative Proof
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Nov 30th 2009, 10:18 PM
  5. Hyperbolic Function Proof
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Nov 10th 2008, 12:24 AM

Search Tags


/mathhelpforum @mathhelpforum