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Math Help - how to solve the following using Euler identity .

  1. #1
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    how to solve the following using Euler identity .

    show that

    exp (i (x + y)) = exp (ix) exp (iy)
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    Re: how to solve the following using Euler identity .

    Euler's identity is:

    e^{\theta i}=\cos(\theta)+i\sin(\theta) hence:

    e^{(x+y)i}=\cos(x+y)+i\sin(x+y)

    Now, use the angle-sum identities for sine and cosine and see if you can arrange the terms to get the desired result.
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    Re: how to solve the following using Euler identity .

    Quote Originally Posted by MarkFL2 View Post
    Euler's identity is:

    e^{\theta i}=\cos(\theta)+i\sin(\theta) hence:

    e^{(x+y)i}=\cos(x+y)+i\sin(x+y)

    Now, use the angle-sum identities for sine and cosine and see if you can arrange the terms to get the desired result.
    thanks for your reply. i ended up with this :
    exp(i(x+y))= cos(x+y)+isin(x+y)
    = cos(x)+cos(y)+isin(x)+isin(y)
    =[cos(x)+isin(x)]+[cos(y)+isin(y)]
    where cos(x)+isin(x)= exp(i(x))
    after substitution we get
    =exp(i(x))+exp(i(y))

    the confusing part is the "+" sign that i'm getting it should be " * "

    any suggestions..
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: how to solve the following using Euler identity .

    You need to use the identities:

    \cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)

    \sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)

    These are the angle-sum identities for sine and cosine.
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    Re: how to solve the following using Euler identity .

    = cos(x)+cos(y)+isin(x)+isin(y)
    =[cos(x)+isin(x)]+[cos(y)+isin(y)]
    where cos(x)+isin(x)= exp(i(x))
    after substitution we get
    =exp(i(x))+exp(i(y))

    \Quote

    yes i'm using these identities.
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  6. #6
    MHF Contributor MarkFL's Avatar
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    Re: how to solve the following using Euler identity .

    It looks to me like you are using:

    \sin(x+y)=\sin(x)+\sin(y)

    \cos(x+y)=\cos(x)+\cos(y)

    and these are not correct, and will give you the wrong result, as you have found.
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  7. #7
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    Re: how to solve the following using Euler identity .

    please take a look at this
    The Way of Analysis - Robert S. Strichartz - Google Books

    I'm using the same method but my answer is ... exp(i(x))+exp(i(y)) which is not correct
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  8. #8
    MHF Contributor MarkFL's Avatar
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    Re: how to solve the following using Euler identity .

    Access is restricted to that page.

    Use the correct identities I gave above, and you will get the correct result. If you get stuck, I will be glad to help.
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  9. #9
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    Re: how to solve the following using Euler identity .

    thank you for your help i got it

    cos(x+y)=cosx.cosy-sinx.siny

    sin(x+y)=sinx.cosy+cosx.siny

    Now, cos(x+y)+isin(x+y)

    =(cosx.siny-sinx.cosy)+i(sinx.cosy+cosx.sinx)

    = cosx.siny-sinx.cosy+isinx.cosy+icosx.sinx

    =(cosx+isinx)(cosy+isiny)[i^2=-1]
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  10. #10
    MHF Contributor MarkFL's Avatar
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    Re: how to solve the following using Euler identity .

    Yes, good work!
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