Deriving Σ sin n using Euler's formula

[DPMachine]

Homework Statement

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:

$$\sum^{\infty}_{n=1} sin(n) = sin(1) + ... + sin(n) = Im(e^{i(1)}+ ... +e^{i(n)}) ...$$

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

$$= Im(e^{i(\frac{n+1}{2})}\frac{sin(n/2)}{sin(1/2)}) = \frac{sin((n+1)/2)sin(n/2))}{sin(1/2)}$$

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?

Homework Equations

The Attempt at a Solution

 

[Mark44]

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

 

[DPMachine]

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