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Deriving Σ sin n using Euler's formula

  1. Oct 22, 2009 #1
    1. The problem statement, all variables and given/known data

    I was looking over my notes and there was a part that didn't make sense.

    It's basically using the Euler's formula ([tex]e^{ix}=cos(x)+isin(x)[/tex]) and the fact that [tex]sin(x)=Im(e^{ix})[/tex] to find what Σ sin n sums to.

    It starts out like this:

    But this part isn't relevant to my question... I'll just skip over to the part that confused me:



    Here, I don't understand how [tex]Im(e^{i(\frac{n+1}{2})}\frac{sin(n/2)}{sin(1/2)})[/tex]

    turned into [tex]\frac{sin((n+1)/2)sin(n/2)}{sin(1/2)}[/tex]

    I understand that [tex]e^{i(\frac{n+1}{2}) = cos((n+1)/2) + isin((n+1)/2)[/tex], by just applying the Euler's formula, but I still can't seem to demystify it. What does the "Im" part do?

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 22, 2009 #2

    Mark44

    Staff: Mentor

    It gives the imaginary part. E.g, if z = x + iy, Im(z) = y and Re(z) = x.
     
  4. Oct 22, 2009 #3
    Oh wow... okay. I don't know why that was so hard to figure out. Thank you!
     
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