# Complex Numbers and amplitude

mmmboh
Hi this isn't homework, just a practice problem I already have the answer too for my waves class:
z=sin(wt)+cos(wt)
Express this in the from Z=Re[Aej(wt+$$\alpha$$)]

I know how to express sine in the form of cosine, and cosine in the from of a complex exponential, but I don't know how to do this...I need to find the amplitude and $$\alpha$$. Can someone help?

## Answers and Replies

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Expand $Re[Ae^{j(\omega t+\alpha)}]$ in terms of sin(ωt) and cos(ωt) and compare it to z.

mmmboh
Well for $Re[Ae^{j(\omega t+\alpha)}]$...the inside equals$Ae^{j(\omega t)}e^{j\alpha}[/tex] and [itex]e^{j\omega t}$=cos(wt)...I'm not really sure where to go from there.

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You need to get the inside into the form x+iy so you can just pick off the x when you take the real part. Don't break the exponential up. Just use Euler's formula on it.

mmmboh
Ok so cos(wt)+sin(wt)=Re[Acos(wt+a)]+Re(Ajsin(wt+a)....and now..I don't really get what the Re does, I know that means real, but what is the significance of it here? Am I suppose to take out the j or something?

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You throw away the imaginary part: Re[x+iy]=x.

mmmboh
Oh I got it thanks!