# Derivation Trouble

1. Sep 6, 2004

### Emc2brain

Derivation Trouble!!!!!

Please read the attachment with this posting: Here's the problem, I have been attempting to derive this for a couple of days now... However, it seems that whatever I do all that I end up deriving is itself again; meaning I get back to where I started. Can anyone give me a few pointers because I'm flat out of luck here. Here I use Euler's Relationships...but no help....hmm..?

e^(i*theta)-1=2i*sin(theta/2)*e^(i*(theta/2))

I have the pretty version in the attachment. :grumpy: :rofl: :grumpy:

Crazed Hannah

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2. Sep 6, 2004

### Emc2brain

liz this is how one posts

3. Sep 6, 2004

### Tide

Note that $1=e^{2\pi i}$ and $a^2-b^2 = (a-b)(a+b)$

4. Sep 6, 2004

### mathman

I will use x for theta.

eix-1=cosx+isinx-1
=cos2x/2-sin2x/2+2icosx/2sinx/2-1
=-2sin2x/2+2icosx/2sinx/2
=2isinx/2(cosx/2+isinx/2)
=2isinx/2eix/2

5. Sep 6, 2004

### Emc2brain

I'm still a little confused on how you got the half angle in there? I seemed to have missed a step

Hannah

6. Sep 6, 2004

### Emc2brain

Because I know that e^(2ix) = cos2x/2-sin2x/2+2icosx/2sinx/2. Where'd you get e^(2ix), because all that I see is e^(ix)? Or is it emplied that 1=e^(2ix)? If that is true then why does your answer still contain a -1? Looking like this: cos2x/2-sin2x/2+2icosx/2sinx/2 -1?

Hannah

7. Sep 6, 2004

### Tide

Even simpler:

$$e^{i \theta} - 1 = e^{i \frac{\theta}{2}} \left(e^{i \frac{\theta}{2}} - e^{-i \frac{\theta}{2}}\right)$$

It's just factoring.

8. Sep 7, 2004

### Emc2brain

Thanx so much....