# Deriving Σ sin n using Euler's formula

1. Oct 22, 2009

### DPMachine

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 ($$e^{ix}=cos(x)+isin(x)$$) and the fact that $$sin(x)=Im(e^{ix})$$ 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 $$Im(e^{i(\frac{n+1}{2})}\frac{sin(n/2)}{sin(1/2)})$$

turned into $$\frac{sin((n+1)/2)sin(n/2)}{sin(1/2)}$$

I understand that $$e^{i(\frac{n+1}{2}) = cos((n+1)/2) + isin((n+1)/2)$$, 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. Oct 22, 2009

### Staff: Mentor

It gives the imaginary part. E.g, if z = x + iy, Im(z) = y and Re(z) = x.

3. Oct 22, 2009

### DPMachine

Oh wow... okay. I don't know why that was so hard to figure out. Thank you!