This meta-query focuses on my (user2661923) answer to the following link, Solving $$3\cos x+4\sin x = \frac52$$, as well as the subsequent comment that my answer triggered.

I encountered a mathSE query that someone else had already provided an answer to. I both liked the already provided answer and felt that I could not improve upon it. At the same time, I felt that there were still questions that needed to be addressed; that is, I felt that it would be beneficial to the OP to address these questions.

This purpose of this meta-query is to give mathSE reviewers the opportunity to express their opinions; in the circumstances that I encountered, was my answer to the original mathSE query appropriate?

If most of you feel that it was not appropriate, then, in future similar situations, I will not post such an answer. Alternative, if you feel that my answer was appropriate, then I will continue to post such answers.

By the way; some of the reactions to this meta-query have convinced me that both the title of the meta-query and the meta-query itself should be revised. Therefore, I've revised the meta-query + title.

I infer from some of the reactions to this meta-query, that my response to the original mathSE query would have been more appropriate as a comment rather than an answer. I agree. Despite that, I posted my response as an answer, simply because I thought it would make my response significantly more legible.

Edit

Recent comments discuss the use of a blog:

I have mixed feelings about relating it specifically to this meta-query.
On the one hand:

Questions on main usually lead to objective, clear, closed-ended answers. ...Expository articles, perhaps detailing how to approach a style of problems, ..., belong on the blog.

On the other hand:

How long does it take for an article to be published after it's written? ... the queue is about one month (re, as of 7-25-2014).

In the underlying mathSE query, Yves Daoust' criticizing comment was upvoted to 5.
In this meta-query:
postmortes' 1st comment was upvoted to 8.
The meta-query itself was first upvoted to 2 and then downvoted back to 0.

Although some reviewers approved of my answer to the underlying mathSE query, it seems to me that most did not. I will be guided by this majority view.

• I think your should make clear in your answer which Ramanujan's analysis are you referring to. The other Ramanujan that I know is famous for proving things in lost book. – Arctic Char Aug 25 '20 at 7:36
• @ArcticChar Reasonable point. However, I don't think that this is why Yves Daoust was objecting. – user2661923 Aug 25 '20 at 7:41
• Your "answer" doesn't actually answer the question asked, but rather discusses it. Since SE is specifically a Q&A site, that's a problem. That said, providing context to an answer is, for me, important, so I don't want to discourage you; rather encourage you to be more focused in how you provide the context -- and not forget to answer the question as well :) – postmortes Aug 25 '20 at 7:41
• There are currently five answers on that question, none of which are substantially different (and none of which make the key observation that given a question involving $3,4$ and $5$ and $\sin$ and $\cos$ the solver's first thought should be right angled triangles). Even if your answer isn't "better" by your evaluation, it needs to be there. You can say that someone else's version is better... but then, why are you writing an answer at all in that case? What you're adding is context, and that is worthwhile -- so either edit the other answer (ask first) to improve it, or write your own – postmortes Aug 25 '20 at 7:51
• To answer the abstract question you ask (and not pass judgement on the answer you give in the linked post): I would rather people took the time to explain their reasoning than they just spat out an answer. I think Math.SE is an appropriate place to "teach", so including "non-analytical methods designed to expand the OP's intuition and facilitate the OP's confronting questions similar to the query" is a good thing. I often get the feeling that people treat Math.SE like it's some sort of weird competition, and that these people don't think about the person who is asking the question. – user1729 Aug 25 '20 at 9:33
• (Incidentally, regarding the comment to your answer: "Mathematics Educators" is a "site for those involved in...teaching mathematics" and not for actual teaching.) – user1729 Aug 25 '20 at 9:37
• @user2661923 Okay, great. It is a more coherent question now. – user1729 Aug 25 '20 at 15:14
• I notice that the OP was told to divide by 5, made one comment about that, but never reached $\frac{3}{5} \cos x + \frac{4}{5} \sin x$ and, well, ran away. – Will Jagy Aug 25 '20 at 16:16
• @WillJagy I confess that I really liked Ramanujan's answer to the mathSE query and based my reaction exclusively on his answer. I'm unsure whether it's because the other answers didn't exist at the time of my original reaction to Ramunajan's answer. If I encountered the original mathSE query now, with all of the reactions that it triggered, I would be reluctant to give a "comment/answer" that ignores everything but Ramunajan's answer. – user2661923 Aug 25 '20 at 16:36
• For those who can't see the question and answers, it might be worth mentioning that the question was closed as "Not meeting MathSE guidelines" and subsequently deleted; I'm pretty sure this is unrelated to user2661923's answer. – Arnaud D. Aug 26 '20 at 13:58
• @AnindyaPrithvi The question was a poor question; it stated the problem and nothing more. If you feel that the answers should be saved then you could repost the question, following the guidelines here. – user1729 Aug 26 '20 at 14:13
• @AnindyaPrithvi I just requested undeletion by flagging the post for moderator intervention. FWIW: I suspect that the closing/deleting of the question is unrelated to my (user2661923) answer to the question. Personally, except for the fact that deleting the original mathSE query may prevent people from having an informed opinion on this (meta) query, I personally support closing/deleting the original mathSE query. The query showed no work and showed no background. The query OP disregarded the opportunity to repair the query. What else was mathSE supposed to do? – user2661923 Aug 26 '20 at 16:03
• @AnindyaPrithvi see here ( math.meta.stackexchange.com/questions/13903/community-blog-faq ) for the history of the math.se blog – postmortes Aug 26 '20 at 19:33
• "I just requested undeletion by flagging the post for moderator intervention." That's not the way to request undeletion. It is, in general, not up to moderators to act to undelete posts. "I personally support closing/deleting the original mathSE query." Now I'm confused. If you support deletion, why do you request undeletion? – Gerry Myerson Aug 26 '20 at 23:27

This is a pseudo-answer intended to provide the original query, Ramanujan's answer, and then my answer. The idea is to allow reviewers of this meta-query to make an informed decision whether or not my answer was appropriate.

%%%%%%%%%

...........
Original Query

Find real solutions of
$$3 \cos(x) + 4\sin(x) = \frac{5}{2}.$$

...........

I just rechecked my link to the original mathSE query. From my perspective, Ramanujan's answer is (still) gone.

...........

I completely agree with Ramanujan's analysis. I also think his stopping point was well timed for the OP to complete the problem. I would like to share underlying ideas with the OP that will help him in future similar situations.

First of all, I think that it is worth demonstrating that
$$t = \tan\left(\frac{x}{2}\right) \;\Rightarrow\; \cos(x) = \frac{1 - t^2}{t^2 + 1}.$$

Use
(1) $$\tan^2(\theta) + 1 = \sec^2(\theta).$$
(2) $$\cos^2(\theta) - \sin^2(\theta) = \cos(2\theta).$$

Since $$\sin^2(\theta) = 1 - \cos^2(\theta),\;$$ (2) above can be re-expressed as
$$2\cos^2(\theta) - 1 = \cos(2\theta).$$

Letting $$\frac{x}{2} = \theta,$$ you have
$$2\cos^2(x/2) - 1 = \cos(x).$$

This implies that $$\frac{\cos(x) + 1}{2} = \cos^2(x/2),$$ which implies that
$$\frac{1}{\cos^2(x/2)} = \frac{2}{\cos(x) + 1}.$$

By assumption, $$t = \tan\left(\frac{x}{2}\right).$$
Using (1) above, this means that $$t^2 + 1 = \frac{1}{\cos^2(x/2)}.$$
This means that $$t^2 + 1 = \frac{2}{\cos(x) + 1}$$, which leads directly to the desired result.

...

(a)
What is the source of this problem, a book, a class, or a contest? If not a contest, then what are the theorems or (previously) solved problems that led up to this problem?

(b)
What "baby steps" might the student take to attack such a problem? What type of exploration?

(c)
Assume that the problem is not from a contest. Also assume that the student has made a reasonable attempt to attack questions (a) and (b) above. This means that the student has meta-cheated, trying to figure out what recent concepts (i.e. tools) from his book or class are relevant, and the student has then taken small exploratory steps to attack the problem using these tools.

When posing this question on mathSE, the OP should share all of his analysis. That is, include the tools that the OP thinks might be pertinent, and give full details on what the OP has tried, and where the OP is having trouble.

(d)
Now consider Ramanujan's analysis, which leads directly to a solution. As shown near the start of my answer, the demonstration that
$$t = \tan\left(\frac{x}{2}\right) \;\Rightarrow\; \cos(x) = \frac{1 - t^2}{t^2 + 1}$$
involves some complicated analysis.

Further, this begs the question: How was the OP (i.e. the student) supposed to know that the substitution $$t = \tan(x/2)$$ would lead to a solution?

...

If this problem is from a contest, then the atmosphere is obviously "all bets are off", and it's (very) fair game that the solution requires unpredictable creativity.

To the OP: if this is the case, then you can ignore the rest of this answer.

If this problem is (instead) from a book or a class, then have you (the OP) been exposed to recent book/class problems that required a substitution like $$t = \tan(x/2)$$?

Further, have you been exposed to problems/theorems that utilized analysis similar to the analysis that I shared that demonstrated that
$$t = \tan\left(\frac{x}{2}\right) \;\Rightarrow\; \cos(x) = \frac{1 - t^2}{t^2 + 1}.$$

In other words, is the solution based on Ramanujan's analysis the solution that your teacher or book intended the student to derive? If not, then what "baby steps" was the student supposed to take that would have (eventually) solved the problem?

The point of my long-winded answer is:
(e) How should the student attack a book/class problem?
(f) How should the student present (any) problem on mathSE.