Periodic functions like sin(x) are the solutions to differential equations like dy/dx = -y that describe for example, oscillations of springs, to name but one of an extremely large number of dynamical systems that behave this way.

It's completely valid and equivalent to formulate trigonometric functions as a dynamic system (x,y-coordinates as you go around the circle using a rotation matrix).

It sounds like you're trying to say something different than the comment you replied to. sin(x) for a range of x is trivial to draw by looking at earlier parts of the curve to determine what to draw next. Back in the slow 1980's home computer days, on machines without floating point and multiply/divide, we quickly got used to approximate sine waves numerically by simply typing out sequences of integers with a rough idea of the wave it'd produce. If you gave me a subset of such a sequence and asked me to complete it, I wouldn't need much of it to assume you were wanting to approximate sin.

Edit: should not be approximated