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Sinusoid to exponential w/ complex frequencies

  1. Jan 16, 2008 #1
    1. The problem statement, all variables and given/known data
    Express the given sinusoid in terms of exponentials and determine the complex frequency(s).

    4(e^-2)cos(wt + pi/4)


    2. Relevant equations

    Euler's identity:

    A(e^i*theta) = A[cos(theta) + i*sin(theta)]


    3. The attempt at a solution

    I'm really drawing a blank here. I tried writing in a "+ i*sin(wt + pi/4)" after the cosine but that really didn't help my any. I'm not even sure what type of math this is considered... please any starting point would be appreciated
     
  2. jcsd
  3. Jan 16, 2008 #2
    Isn't there an expression for cos (z) in terms of complex exponentials? Check your text and try to use it. That seems to be what they are asking for.
     
  4. Jan 16, 2008 #3
    I found this one online:

    cos(theta) = (1/2)[e^(+i*theta) + e^(-i*theta)]

    applying this to mine I get:

    (4e^-2){(1/2)[e^(+i*(wt + 45)) + e^(-i*(wt + 45))]

    rewrite as:

    ((4e^-2)/2)[(e^iwt)(e^i*45) + (e^-iwt)(e^-i*45)]

    I guess that's my answer...but what about complex frequencies??
     
  5. Jan 16, 2008 #4
    First off, change those 45s back to pi/4s. Too much of this stuff is invalid in degrees - always using radians when working in trig/exponential functions is a good habit to get into, unless you're specifically told not to use them (which will likely never happen).

    If I'm remembering my Complex Analysis course correctly, the complex exponential function is periodic with period 2(Pi)i (that last i is i^2=-1, that i). Frequency generally refers to the number of cycles per standard period, but if you could give us the definition that you have it would be helpful.

    It's likely that you could start from the original form and use the fact that cos(wt + pi/4)
    will undergo one cycle as (wt + pi/4) ranges from 0 to 2Pi, but someone may certainly come in and correct me.
     
    Last edited: Jan 16, 2008
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