What is the Complex Exponential Form of a Trigonometric Function?

[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[Ae^j(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?

 

[vela]

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 equalsAe^{j(\omega t)}e^{j\alpha}[/tex] and e^{j\omega t}=cos(wt)...I'm not really sure where to go from there.

 

[vela]

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?

 

[vela]

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

 

[mmmboh]

Oh I got it thanks!