Euler's Relationship! Need Help!

[Emc2brain]

if e^(i*theta) = cos(theta) + i*sin(theta)

then what is e^(-2i*theta) = ????

I attempted to derive this and got the following for the +2i:
e^(+2*theta) = cos(2*theta) + 2i*cos(theta)sin(theta)

Not even sure if this may be correct, but I believe the answer to my question with negative 2 (-2i) must be simple... Help please, thanx.


Because I am attempting to derive 2sin^2(theta) = 1-cos(2theta) from euler's relationship: e^(i*theta) = cos(theta) + i*sin(theta)


Hannah
:blushing:

[chroot]

Euler's relation is that

e^{ix} = \cos(x) + i \sin(x)

where x can be anything at all. In your example, x would be -2 \theta, so plug it in:

e^{-2 i \theta} = \cos(-2 \theta) + i \sin(-2 \theta)

- Warren

[Emc2brain]

Thanks, that helps!

[Emc2brain]

Just solved it, after 45 minutes... :frown:

[Tide]

Euler's relation is that

e^{ix} = \cos(x) + i \sin(x)

where x can be anything at all. In your example, x would be -2 \theta, so plug it in:

e^{-2 i \theta} = \cos(-2 \theta) + i \sin(-2 \theta)

- Warren
And e^{-2i\theta}[/tex] is also [itex]\left(e^{-i\theta}\right)^2 which gives \cos^2\theta-\sin^2\theta-2i\sin\theta\cos\theta. :smile:

[HallsofIvy]

Since you arrived at e^(+2*theta) = cos(2*theta) + 2i*cos(theta)sin(theta)
I'm surprised you could continue: using -θ instead of θ just replaces θ with -θ and cos(-θ)= cos(θ), sin(-θ)= -sin(θ).

Also, since you clearly replaced sin(2θ) with 2sin(θ)cos(&theta), why not also replace cos(2&theta) with cos2(θ)- sin2(θ)?

Putting those together, e^{-2\theta}= cos(-2\theta)+ i sin(-2\theta)
= cos^2(-\theta)- sin^2(-\theta)+ 2i sin(-\theta)cos(-\theta)
= cos^(\theta)+ sin^2(\theta)- 2i sin(\theta)cos(\theta),
exactly what Tide got by squaring.

[Emc2brain]

THANK YOU SO MUCH GUYS... you've all been too helpful :blushing:

Hannah

[Emc2brain]

Hello there helpful bunch! ;)

How are you guys able to write out the equations?? Because I tried to copy and past them into this email however it simply would not do that...Thanx for all the assistance!!

[chroot]

The equations are written with in a language called TeX, which our forum software parses and turns into images.

See more here: http://www.physicsforums.com/showthread.php?t=8997

- Warren