\(Re\left \{ b(t) \right \}\) where \(b(t)\) is an imaginary number,

\(b(t)=Ae^{\mathit{j (\omega t+ \phi)}}=Ae^{\mathit{j \phi}}e^{\mathit{j \omega t}}\). Now, the real part of \(\widehat{A}=Ae^{\textit{j} \phi}\) multiplied by \(cos(\omega t)\) gives us the original \(a(t)\) back. I.e. \(Re\left \{ \widehat{A} \right \}=Acos(\phi);

a(t)=Acos(\phi)*cos(\omega t)\)

Moreover, looking at the KFUPM Open Courseware I also found that a phasor should have a j multiplied in the exponent. So, what is the proper representation? Whom should I believe and what exactly should I leave in my notes?