Although with many recent proofs and methods about evaluation of values of $\sin 18^\circ$, $\cos 54^\circ$, etc, it is said that ancient Indian mathematics knew about the proper value of such basic trigonometric values.
How did they determine such values? How does the method work on a given special angle?
These are angles that we should expect to be known in antiquity. They would have been known to the ancient Greeks, for example, since they are explicitly constructable.
To construct $54^\circ$, first construct a perfect pentagon (which can be done very simply) and then bisect any inner angle.
To construct $18^\circ$, note that the exterior angle of a pentagon has angle $72^\circ$. Bisecting this angle twice gives $18^\circ$.
Thus understanding these angles (and their corresponding sine and cosine values) is about as difficult as understanding a pentagon and understand what happens under angle bisection. Both the pentagon and bisection are very classical, and so we should expect most mathematically inclined civilizations to have understood these angles and values.