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)
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 I am vs Re and successfully convert to polar coordinates. Help?