Is there any simple formula or algorithm for determining if a point lies above or below the sine curve? For instance, if I have a point $(x, y)$, how can I test whether or not $y > \sin(x)$? Obviously taking the actual $\sin(x)$ (or $\cos(x)$) is not an option otherwise I wouldn't be asking.
All three angles, $A, B$, and $x$ are first-quadrant angles in $[0, 90°]$.
Additionally, I know two reference points, A and B such that A < x < B, and I know both the sine and cosine of A and B. I thought perhaps comparing slopes might be useful, for instance, I know that the slope of the tangent at A is greater than the slope of the secant from A to x, which is greater than the slope of the secant from x to B, which is greater than the slope of the tangest at B. But I haven't been able to come up with a way to actually use any of that.
To clarify what I'm after: I'm working on doing rapid estimations of various useful functions like sines, cosines, exponents, logs, etc., for the many cases that arise in which fast, approximate answers are useful (e.g., assumption checking during debugging or feasibility evaluation).
I'm currently able to estimate sines and cosines of any angle in degrees to within $10\%$ error, but I'd really like to be able to take those initial estimates and then refine them further with some kind of simple iterative process that can be carried out relatively quickly with pencil and paper. I find this useful in various situations, for instance in the lab or in group brainstorming sessions, in which a calculator is not readily available.