this is simple of wave function:
\Psi{(\vec r, t)}=Ae^{i(\phi+\vec k .\vec r - \omega t)}

\Psi{(\vec r, t)}=A|\cos{(\phi+\vec k .\vec r - \omega t)}+i\sin{(\phi+\vec k .\vec r - \omega t)}|
i want to know
why wave function\Psi{(\vec r, t)} shown with \vec randt?

what is A Amplitude?

what is phase angle (\phi+\vec k .\vec r - \omega t)?

and what means cos and sin of it \cos{(\phi+\vec k .\vec r - \omega t)}?

i want to know why wave function\Psi{(\vec r, t)} shown with \vec randt?


Because the \Psi is a function both of position \vec r and time 't'.



what is A Amplitude?

Yes , 'A' typically is used to mean "amplitude."



what is phase angle (\phi+\vec k .\vec r - \omega t)?

The phase angle defines how the magnitude of the wave is shifted relative to \sin (\omega t). So here the phase angle is \phi + \vec k \cdot \vec r.



and what means cos and sin of it \cos{(\phi+\vec k .\vec r - \omega t)}?

This comes from:


e^{i \theta} = \cos(\theta) + i \sin (\theta)