# Express the following in the form of a Complex Number

## 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$\alpha$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 $\alpha$ = 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?

##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}##.

rude man
Homework Helper
Gold Member
How about changing your x into x = Acos(x + ψ)? Just high school trig.

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so i made the e^i$\pi$/4 into cos($\pi$/4)+sin($\pi$/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?

so i made the e^i$\pi$/4 into cos($\pi$/4)+sin($\pi$/4) by using eulers formula. Now i calculated the complex amplitude to be √2.414.

This is the complex amplitude of what?