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I am going around in circles, excuse the pun, with phasors, complex exponentials, I&Q and polar form...

1. A cos (ωt+Φ) = Acos(Φ) cos(ωt) - Asin(Φ)sin(ωt)

Right hand side is polar form .... left hand side is in cartesian (rectangular) form via a trignometric identity?

2. But then sometimes I read...

A cos (ωt+Φ) in polar form has corresponding cartesian form of Bcos(ωt)+Csin(ωt), which is fine to understand because this cartesian form gives X and Y coordinates on a cartesian coordinate axes of a vector in that axes.

3. But point 1 and 2 are different, how can Acos (ωt+Φ) in polar represent Bcos(ωt)+Csin(ωt) in cartesian but also be equal to Acos(Φ) cos(ωt) - Asin(Φ)sin(ωt) via a trignometric identity ---> Is it because Acos(Φ) and Asin(Φ) are constants and therefore also B and C? Might be obvious but I need to ask for my own sanity of seeing so much different ways its written.

What about if B and C are not constants due to the phase changing with time Φ(t)?

I am further purplexed by notation used for complex sinusoids.

3. Acos (ωt+Φ) can be represented as the real part of Ae^{i(ωt+Φ)}= Acos(ωt+Φ) + iAsin(ωt+Φ)

but from point 1, the right hand side of this equation can be then re-written with the trigometric identity in point 1, expanding it into 4 terms which removes the phase from the argument and giving constants, like in point 3. So why cannot it not be written without the Φ in the argument on the right hand side and use different constants

Ae^{i(ωt+Φ)}=Bcos(ωt)+iCsin(ωt)

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# I Sinusoids as Phasors, Complex Exp, I&Q and Polar form

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