What does "Instantaneous Angle" means
Instantaneous usually means the value of something right at this time. I think what this means in your case is that the function $f(t)$ is complex and you are taking the argument (finding the angle):
$$
\theta(t) = \arg\left( f(t) \right) = {\rm atan}(\Im[f(t)],\Re[f(t)])
$$
where $\Im[ \cdot ]$ indicates taking the imaginary part and $\Re[ \cdot ]$ indicates taking the real part.