Euler's Relationship! Need Help!

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  • #1
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Help! Euler's Relationship!

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:
 
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Answers and Replies

  • #2
chroot
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Euler's relation is that

[tex]e^{ix} = \cos(x) + i \sin(x)[/tex]

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

[tex]e^{-2 i \theta} = \cos(-2 \theta) + i \sin(-2 \theta)[/tex]

- Warren
 
  • #3
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Thanks, that helps!
 
  • #4
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Just solved it, after 45 minutes... :frown:
 
  • #5
Tide
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chroot said:
Euler's relation is that

[tex]e^{ix} = \cos(x) + i \sin(x)[/tex]

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

[tex]e^{-2 i \theta} = \cos(-2 \theta) + i \sin(-2 \theta)[/tex]

- Warren
And [itex]e^{-2i\theta}[/tex] is also [itex]\left(e^{-i\theta}\right)^2[/itex] which gives [itex]\cos^2\theta-\sin^2\theta-2i\sin\theta\cos\theta[/itex]. :smile:
 
  • #6
HallsofIvy
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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, [tex]e^{-2\theta}= cos(-2\theta)+ i sin(-2\theta)[/tex]
[tex]= cos^2(-\theta)- sin^2(-\theta)+ 2i sin(-\theta)cos(-\theta)[/tex]
[tex]= cos^(\theta)+ sin^2(\theta)- 2i sin(\theta)cos(\theta)[/tex],
exactly what Tide got by squaring.
 
  • #7
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THANK YOU SO MUCH GUYS... you've all been too helpful :blushing:

Hannah
 
  • #8
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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!!
 
  • #9
chroot
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