Talk:Sine of Sum/Proof 7

The more I think about it, the more I disagree with the objection that this proof needs to be expanded to cover cases other than the first quadrant. It's perfectly valid for the triangle definition of sine and cosine.

If you want to relate it to the functions like

For: $90 < \theta \le 180$:

$\sin \theta = \sin \tuple {180 \mathop - \theta}$

This belongs somewhere in basic properties, right? Not here. Here, I suppose, under Definition from Circle --Telliott99 (talk) 09:43, 18 October 2023 (UTC)

While this works for the triangle definition of sine and cosine, it remains incomplete. The main theorem, of which this purports to be a proof, is not so limited. --prime mover (talk) 09:57, 18 October 2023 (UTC)

Well it's your site, I guess. But I don't buy it. The basic relationship is from the geometry and this proof works fine. It's crazy that when you want to prove this theorem you start with Euler. What I would ask is how can I state the theorem so that it's restricted to the first quadrant? Grafting the rest of that stuff on here seems inelegant, at least to me. --Telliott99 (talk) 01:20, 19 October 2023 (UTC)

Respectfully disagree. The theorem does not mention limiting the domain of the variable, so the proof should not limit the domain. --Robkahn131 (talk) 01:39, 19 October 2023 (UTC)

So you're saying this is hopeless and doesn't belong here? Unless you can show me the way, I don't see how to adapt this proof to cover the other quadrants. You are looking at the sine as a function over the reals. I think this proof is perfectly fine in the first quadrant. Maybe make a new statement of a restricted theorem so that it expressly applies only to the triangle defintion of sine and cosine? Of course, one can modify Definition:Sine/Definition_from_Circle (notice the /) to state all the cases and the algebra, but that doesn't seem like a proof to me. --Telliott99 (talk) 02:13, 19 October 2023 (UTC)

(One could of course argue that we already have several proofs for Sine of Sum, and pretty though it is to fill us up with yet more proofs of the same thing, these are not essential to the further development of mathematics coverage on $\mathsf{Pr} \infty \mathsf{fWiki}$.)

But here is what I imagine could be done. We have a number of proofs for a number of theorems which do not apply for the entire domain for which the theorem is valid. What one could do is repurpose this proof as a lemma page, in which the domain is indeed limited to the angles that one can construct within a cyclic quadrilateral, and build up a battery of proofs for that lemma. Then one could create another page which fills in (a) the rest of the quadrants, and (b) extends it to the complex plane as necessary.

I found one example using the qualifier "Restricted" but I guess I will leave it to you. I would hope the two proofs can stay. I think they are strikingly simple and pretty proofs (and note that there are similar ones for the difference of angles). --Telliott99 (talk) 12:43, 19 October 2023 (UTC)

I note the same applies to Proof 6 as well, in which a similar exercise needs to be done. --prime mover (talk) 04:58, 19 October 2023 (UTC)