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Homework Help: Euler's Formula

  1. Feb 20, 2008 #1
    1. The problem statement, all variables and given/known data




    x(t)=(exp^(-[tex]\zeta[/tex][tex]\omega[/tex]t))*(A sin ([tex]\omega[/tex]*t + [tex]\varphi[/tex]))

    2. Relevant equations


    3. The attempt at a solution

    I've managed to get x(t) = (exp^(-[tex]\zeta[/tex][tex]\omega[/tex]t))*((a1(cos[tex]\omega[/tex]*t) + i sin ([tex]\omega[/tex]*t))+(a2(cos[tex]\omega[/tex]*t) - i sin ([tex]\omega[/tex]*t))) then when i simplify things...sin terms cancel out and i end up geting...

    exponential term * (a1+a2) * (2cos [tex]\omega[/tex]*t)
  2. jcsd
  3. Feb 20, 2008 #2

    Tom Mattson

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    Is the [itex]\zeta[/itex] a constant here?

    No, the sine terms do not cancel out. You have the following.

    [tex]x(t)=e^{-\zeta\omega t}\left[(a_1+a_2)\cos\left(\omega\sqrt{1-\zeta^2}t\right)+(a_1-a_2)\sin\left(\omega\sqrt{1-\zeta^2}t\right)\right][/tex].

    The only way the sine term will vanish is if [itex]a_1=a_2[/itex], and you did not say that that is the case in your problem statement.
  4. Feb 20, 2008 #3
    Hello. It is a constant.
    Yeah, I figured out that I made a mistake.

    that's as far as i can go but i need to combine that into sin like i've posted above.

    thanks for the reply!

    And, there should be a bracket infront of "sqrt" so exponential part it outside the whole thing.

    [edit] I think I got it...Thank you very much! :)
    Last edited: Feb 20, 2008
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