- #1

Mentallic

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## Homework Statement

Show that if [tex]\theta[/tex] is not a multiple of [tex]2\pi[/tex] then

[tex]Im\left(\frac{1-e^{i(n+1)\theta}}{1-e^{i\theta}}\right)=\frac{sin\left(\frac{1}{2}(n+1)\theta\right)sin(\frac{1}{2}n\theta)}{sin\frac{1}{2}\theta}[/tex]

## Homework Equations

[tex]e^{i\theta}=cos\theta+isin\theta[/tex]

## The Attempt at a Solution

I noticed that [tex]\frac{1-e^{i(n+1)\theta}}{1-e^{i\theta}}\right)[/tex] is a geometric summation with [tex]e^{i\theta}=r[/tex] then we have:

[tex]1+e^{i\theta}+e^{i2\theta}+...+e^{in\theta}[/tex]

So,

[tex]Im\left(1+e^{i\theta}+e^{i2\theta}+...+e^{in\theta}\right)=sin\theta+sin2\theta+...+sin(n\theta)[/tex]

I have no idea how to show this summation is equal to what I have to show. Most likely I'm not even headed in the right direction.