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Complex Analysis

  1. Jan 12, 2010 #1
    1. The problem statement, all variables and given/known data

    By considering the real and imaginary parts of the product eithetaeiphi, prove the standard formulae for cos(theta+phi) and sin(theta+phi)



    2. Relevant equations

    The standard formula for:
    cos(theta+phi) = cos(theta)cos(phi) - sin(theta)sin(phi)
    sin(theta+phi) = sin(theta)cos(phi) + sin(phi)cos(theta)

    3. The attempt at a solution

    from hereon out, lets let t=theta and p=phi. I can't figure out how to put the symbols in there on this site

    ei(t+p) = [cos(t) + isin(t)] [cos(p) + isin(p)]

    = cos(t)cos(p) - sin(t)sin(p) + isin(t)cos(p) + isin(p)cos(t)

    REAL PART = cos(t)cos(p) - sin(t)sin(p)
    IMAGINARY PART = isin(t)cos(p) + isin(p)cos(t)

    but I can't use the trig identities because I have to prove them. I'm probably not going in the right direction, but if somebody could point me in the right way, that would be great!

    Thanks!
    Andrew
     
  2. jcsd
  3. Jan 12, 2010 #2

    Matterwave

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    Gold Member

    You could then expand [tex]e^{i(\phi+\theta)}=cos(\phi+\theta)+isin(\phi+\theta)[/tex] by just considering phi+theta to be collectively x in the Euler's formula. If you match the real and imaginary parts you should get the correct trig identities.
     
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