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Phasors
Handling the trigonometric functions can be tedious:
This is just a simple phase shift, say,  = t,  = 0.
We have Euler's identity:
that relates trigonometric functions to complex exponentials,
the math of which is a bit simpler:
e j (    )  e j e j
At least, phase shifting is much easier!
So, we use ej to represent cos for mathematical convenience.
But, why can we do this? What’s behind this?
Based on this, we have
(say, a voltage)
This rotation part is always there.
So leave it out.
(or "complex amplitude)
leave out.
In both cases, the time dependence is left out, since we are dealing
with a single frequency.