I have a know-how and a library about trigonometry and trigonometric operations, I would like to know if I can possibly rely on trigonometry for parametric curves too and how the trigonometry from the circle with $\mathbb{ \mathcal{r} = 1\ unit}$ .
My interest is mostly about measuring angles, distances and all the good stuff that trigonometric functions and properties reveal, like with atan2 in C/C++ that given 2 points computes the angle between them ( clockwise ), and what kind of properties I get if apply the same rules to a parametric curve ?
To be honest by parametric curve I mean bezier curve because that's what I have in input like 99.9% of the time ( note that with bezier I mean any kind of bezier, mostly cubic and quadratic ) but I would like to know the general idea about applying trigonometry to a parametric curve ( if this is possible ).
A few examples ...
If the vector $U = (u_x, u_y)$ is the derivative vector of a Bézier curve (or any other curve, actually), then $\text{atan2}(u_x,u_y)$ is the angle between the curve tangent and the $x$-axis.
Once you know to calculate this angle, you can find places where the tangent is horizontal or vertical, which is often useful.
If you calculate this angle at two points that are very close together, you can get an estimate of the curvature of the curve.
You can measure the total angle that the curve turns through, which sometimes indicates that it needs to be split in order for some computation to work reliably.
Since Bézier curves are parameterized using polynomials, trig functions do not come into the picture very much (less so than with circles, for example).
It sounds like your trig function software/expertise is a so-called "solution in search of a problem".