Express the following in the form of a Complex Number

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Homework Statement


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)


Homework Equations



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

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?
 

Answers and Replies

  • #2
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##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}##.
 
  • #3
rude man
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How about changing your x into x = Acos(x + ψ)? Just high school trig.
 
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  • #4
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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?
 
  • #5
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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.
This is the complex amplitude of what?
 
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