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Express the following in the form of a Complex Number

  1. Sep 18, 2013 #1
    1. The problem statement, all variables and given/known data
    For my waves class, I have to do this problem. I've previously completed a question like this except there was no phase constant (∏/4) in that question.
    Express the following in the form x = Re [Ae^i[itex]\alpha[/itex]e^iwt
    x=cos(wt + ∏/4) - sin(wt)


    2. Relevant equations

    euler's formula e^iwt = cos(wt) + isin(wt)
    complex amplitude = √A2+B2
    complex angle = tan [itex]\alpha[/itex] = y/x

    3. The attempt at a solution

    I know that cos(wt + ∏/4) = Re [e^i(wt + ∏/4)] and -sin(wt) = Re [ie^iwt]

    combining these two, i have Re [e^i(wt + ∏/4) + ie^iwt] which, after factoring, becomes
    Re [e^iwt (e^i∏/4 + i)]

    My problem is, what do i do with the e^i∏/4 to get the complex amplitude and complex angle? In my previous problem without the ∏/4 shift, i was able to plot in Cartesian coordinates Im vs Re and successfully convert to polar coordinates. Help?
     
  2. jcsd
  3. Sep 18, 2013 #2
    ##e^{i \pi/4}## is a complex number. So is ##e^{i \pi/4} + i##. Find it, then represent it as ## A e^{i\alpha}##.
     
  4. Sep 18, 2013 #3

    rude man

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    How about changing your x into x = Acos(x + ψ)? Just high school trig.
     
    Last edited: Sep 19, 2013
  5. Sep 18, 2013 #4
    so i made the e^i[itex]\pi[/itex]/4 into cos([itex]\pi[/itex]/4)+sin([itex]\pi[/itex]/4) by using eulers formula. Now i calculated the complex amplitude to be √2.414. and the complex angle to be 35.26 degrees or 0.6155 rad. The answer i get is x = Re [√2.414 e^i0.6155t e^iwt ] is this correct?
     
  6. Sep 18, 2013 #5
    This is the complex amplitude of what?
     
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