Timeline for Prove that sin(x) ≥ x/2, but without calculus!
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Sep 11, 2021 at 6:20 | comment | added | justhalf | @Stef, it is mentioned in the question already. | |
| Sep 10, 2021 at 18:51 | comment | added | Noone AtAll | can I ask to repeat the graph, but scaling it to clearly show pi/2? (for example changing all instances of x to pi*x, so that the pi/2 becomes 0.5) | |
| Sep 10, 2021 at 8:52 | comment | added | Stef | "It is obvious because it depends on an assumption which I didn't mention in my whole post" is a complete self-contradiction. If you had simply said "Since we know that 0 <= α <= pi/2, it follows that rectangle ABED fully covers sector BAC", I wouldn't have commented. But saying that a statement is obvious because it is true on a picture, when in fact that statement is false for most values of α, is a very misleading way to write math proofs. In fact, since you don't explicitly use the assumption 0 <= α <= pi/2 in your whole proof, you effectively proved that sin(x) >= x/2 for all values of x. | |
| Sep 10, 2021 at 8:23 | comment | added | justhalf | Agree with Glen here, it is quite obvious. But I guess, Glen, perhaps including "(since alpha is between 0 and pi/2)" would improve the answer by reminding the readers about the constraints of the question? | |
| Sep 10, 2021 at 0:34 | comment | added | Glen O | @Stef - it's obvious because $\alpha$ is known to be between 0 and $\pi/2$. This single piece of information, plus the image provided, allows one to quickly see that constructing the line perpendicular to AB through C forms two lines that bound the sector vertically, and defining points D and E by drawing lines perpendicular to AB through A and B and intersecting with the first line provides lines AD and BE that bound the sector horizontally - as shown in the image (the construction process, that is). To put it another way, you're constructing ABED to cover the sector. | |
| Sep 9, 2021 at 15:05 | comment | added | Stef | "It is obvious that rectangle ABED fully covers sector BAC, and therefore sinα≥α/2" I am wary of any geometric proof that says something is obvious, just because it looks true on one example picture. In particular, this would be false if α wasn't in range [0, pi/2]. So it's really not that obvious at all and in fact requires to be quite careful about the value of α. | |
| Sep 9, 2021 at 11:51 | history | edited | Glen O | CC BY-SA 4.0 | added 1109 characters in body |
| Sep 9, 2021 at 9:22 | comment | added | justhalf | This should be the accepted answer. Haha | |
| Sep 9, 2021 at 6:26 | comment | added | Jaap Scherphuis | Amazing! It's so obvious, so why didn't I see it?! | |
| Sep 9, 2021 at 1:12 | history | edited | Glen O | CC BY-SA 4.0 | Adding spoiler marks to hide answer. |
| Sep 9, 2021 at 0:47 | history | answered | Glen O | CC BY-SA 4.0 |