After when this puzzle was posted, many people pointed out errors and improvements that could be made. I also noticed many flaws, so the post once had gone through drastic changes. However, I learned and understood that such behavior is discouraged. I'll leave the puzzle as it was before the flaws were and edited, to make sure the answers and comments make sense for future readers.
The title explains it all. Can you prove that $\sin(x) \ge x/2 $ holds where $0 \le x \le \pi/2$, but without using any calculus?
The only prerequisite that you can regard as true without proving is that on a plane, if a shape Y can be completely covered by a shape X, then X has a larger area than Y.
- The theorems you incorporate should not have been derived from calculus, so it would be safe to use most formulas of trigonometry.
- The title of this puzzle was inspired by this challenge.
- Hints and my own solution will be posted after some time passes.
- How far can you increase this lower bound, without the aid of calculus?