So I had a quick conceptional question about phasors that I had trouble to understand on my own.
My book establishes this relationship:
And also we know that these forms are equal
Where the right hand side of (9.25) is in the "polar form" of (9.15). So then according to 9.15, its equivalent to the Exponential form:
$$V_m\cos{(\omega t + \phi)} = V_{m} \angle\phi = V_m e^{j\phi}$$ and since
$$e^{j\phi} = \cos{\phi} + j\sin{\phi}$$
doesn't this mean
$$V_m\cos{(\omega t + \phi)} = V_m(\cos{\phi} + j\sin{\phi})$$
So my question is, where is the sine term in the above equation? Since on the RHS we have cosine and sine, but on the LHS we have cosine only. Like I'm just confused on whether the equation is true or not or if I made a mistake. Seems to me the sine is "missing" and when we represent a cosine $$V_m\cos{(\omega t + \phi)}$$ as $$V_m e^{j\omega}$$
it is wrong since the $$e^{j\omega}$$ has both a cosine and sine component to it.