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Complex Trig, DE

  1. May 24, 2013 #1
    1. The problem statement, all variables and given/known data
    Hello,
    I am in differential equations currently and I have a homework question regarding simplifying

    sin( Pi t)/4

    into

    .5 * i E^(-.25 i pi t) - .5 * i E^(.25 i pi t)


    2. Relevant equations
    I think they might be using Euler's Identity, but I am unsure.
    E^(a + ib)t = E^(at) (cos[bt] + i sin[bt])​​
     
  2. jcsd
  3. May 24, 2013 #2

    LCKurtz

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    What happens if you subtract these two equations:$$
    e^{i\theta} = \cos\theta + i\sin\theta$$ $$
    e^{-i\theta} = \cos\theta - i\sin\theta$$
     
  4. May 24, 2013 #3
    I get:

    2 i Sin(theta)

    Even still, how does that get me closer to my end?

    Thank you.
    -James
     
  5. May 24, 2013 #4

    LCKurtz

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    You didn't write the whole equation. When you add/subtract two equations you get another equation.
     
  6. May 24, 2013 #5
    Ok,

    E^(i theta) - E^(-i theta) = 2 i sin(theta)

    I'm still a little bit confused as to where I can go from this.
    I like the way the right hand side of the equation is looking, but I don't know what to do with the imaginary component in 2 i sin(theta)

    Thanks,
    -James
     
  7. May 24, 2013 #6

    LCKurtz

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    Look at what you have and what you are trying to get and what value of ##\theta## you need.
     
  8. May 24, 2013 #7
    Ok, so I have
    .5 * i E^(-.25 i pi t) - .5 * i E^(.25 i pi t)

    knowing:
    E^(i theta) = cos(theta) + i sin(theta)
    E^(-i theta) = cos(theta) - i sin(theta)
    so
    .5 i E^(i theta) = .5 i (cos(theta) + i sin(theta)) = .5 (i cos(theta) - sin(theta))
    .5 i E^(i -theta) = .5 i (cos(theta) - i sin(theta)) = .5 (i cos(theta) + sin(theta))

    I guess we can let theta = .25 pi t
    Can I say,
    -.5 i E^(i theta) + .5 i E^(i -theta)
    = -.5 (i cos(theta) - sin(theta)) + .5 (i cos(theta) + sin(theta))
    = sin(theta)
    = sin(.25 pi t)

    But where does the 1/4 come into play?
     
  9. May 24, 2013 #8
    So sorry! made a big bobo!

    I want to show:
    .5 * i E^(-.25 i pi t) - .5 * i E^(.25 i pi t) = Sin(pi t/4)
    Using the above does just that.

    Thanks so much!!! :)
     
  10. May 24, 2013 #9

    LCKurtz

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    You are making this way too hard. You already have this:

    Now, instead of doing everything all over like you have here:

    Just put your ##\theta = \frac{\pi t}{4}## in that equation you already have at the top of this post:$$
    e^{i\theta}-e^{-i\theta} = 2i\sin\theta$$Lose the decimals, do the substitution, and solve that equation for the sine term.
     
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