Talk:Pythagorean theorem

2391:

no matter where or when the measurements are made, provides inductive support for the homogeneity, isotropy, and scale-invariance of the world. We can deduce the parallel postulate and other important elements of geometry from the Pythagorean theorem. Doesn't this seem like it makes more sense than trying to empirically verify the parallel-postulate? And at very large scales empirical support for the Pythagorean theorem fails, leading naturally to other geometries. Maybe somebody with greater wiki expertise could add a section on reverse mathematics atleast mentioning this perspective. The article introduction asserts that there are many ways to "prove" the Pythagorean theorem, but gives no clear acknowledgement of the parallel postulate or alternatives upon which proofs should be critiqued.

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magnet. That sort of thing is only marginally effective at keeping the cruft out of the main article and instead encourages the accumulation of more cruft. Instead, keeping it only in this one article maintains the pressure to stay at roughly the amount of content that we already have: a properly sourced statement that there are huge numbers of proofs that you can find in certain books, and a small (and I hope carefully-curated) selection of proofs. —

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of Book I of Euclid's Elements should likewise count as "trigonometry", including the Pythagorean theorem itself. By typical definitions of trigonometry, however, the subject involves some relation between lines and circular arclengths or angle measures, and really starts with Hipparchus; centuries-older approaches from Egypt and Mesopotamia are a kind of "proto-trigonometry" at best. –

787:. Any other trigonometric proof must use this foundational principle. All the proofs suggested in this talk page use this foundational principle and some other trigonometric properties. This makes them definively less interesting and less elegant than the proofs that are already there. So, they have a low encyclopedic value and do not deserve to be mentioned. 3073:(infinitely many) proofs of the Pythagorean theorem can't be in scope here, but the proofs can be categorized into 4–5 broad groups, and 1–3 notable examples from each group should be included on this page, irrespective of what material is included on other articles. Many are quite short or can be expressed pictorially. – 3135:

This may be the way to go. I am not familiar with these other ways. For example, I consider Knowledge to be fairly reliable because there are many good editors keeping an eye out for quality; are these other options as reliable in practice and by reputation? Because, if not, I'd like there to be a

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I am inclined against this, on general following-the-sources grounds. In my experience, the texts that cover the Pythagorean theorem at an introductory level don't just apply it; they prove it in one or more ways. We'd be the oddballs if we separated the proofs out entirely. Doing mundane cleanup and

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A couple more notes: Even if the article is split at least 5–6 different proofs should be covered in detail on the main page, taking roughly as much space they currently take. I'm concerned that an explicit article about proofs would become an indiscriminate grab-bag of mediocre crap, and it would be

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I don't have time now, tho maybe i'll do this myself later. (1) All figures should be numbered, and referred to by number in the text, not just in this section but over the entire article. A good way would be to number sections and do Figure 1-1, 1-2, 2-1, etc. so renumbering does not have to occur

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Whether the seqed is part of trigonometry is a semantic dispute rather than a historical/factual one. The seqed is not relevant to the type of "trigonometry" intended when someone says "trigonometric proof of the Pythagorean theorem". By any definition that includes the seqed as "trigonometry", most

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Yes but there is the rub. Knowledge is fairly reliable and you have good editors keeping an eye because we restrict our content. That exactly a reason to avoid long proofs or list of long proves as their verification takes more time/resources and they are less likely to be checked in detail by other

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I can see an interest of (some) readers to have comprehensive collection of proofs, which doesn't fit into this article. But imho Knowledge is not the appropriate place for that, there are other options within in Wikimedia to provide such a collection to readers. One could integrate it into existing

2390:

Seems like everybody has this backwards. The Pythagorian theorem is a generalization of empirical observations, probably going back to ancient monument construction. Observations that all right triangles satisfy the Pythagorean theorem to within precision of the methods available in ancient times,

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This seems like a Mediawiki problem. The infobox is at the top of the page in the source, and the equation immediately follows the paragraph. I think Mediawiki's mobile view perhaps special-cases the leading image or infobox to move it after the first paragraph? Not sure if there's a good workaround

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You claimed it was first further up this page. But the text in the article itself presented no indication that it is noteworthy - Which is why it got deleted. Subjective claims about simplicity and simplifying things on the talk page might be a fun diversion, but the only way a mention could stay in

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Pythagorean theorem dates from more than 1,000 years; trigonometry date from more than 500 years. Since them, hundred of great mathematicians have studied their relationship. So it is very unlikely that something really new can be found on this subject. So, for mentioning Zimba's proof, one requires

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There are literally hundreds of proofs of the theorem, maybe thousands. Picking out and including only one of these, sourced only to its primary publication, makes no sense, because there is no clear selection criterion for it that would not also cause us to also include hundreds of other proofs. We

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I would like to see that split. I think that readers who are looking for multiple proofs can be substantially different from readers who are looking to learn non-proof aspects of the Pythagorean theorem. I think that fully supporting both goals, now and into the future, will make a single article

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The problem with numbering the figures in semi-popular Knowledge articles is that the numbering very rarely stays up to date as many Wikipedians make slight changes here and there. It takes someone constantly checking to maintain the numbering. Per the manual of style, sections " not be numbered or

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a proof or proofs are obviously directly relevant. Indeed I would hope every article about a theorem should include at least some kind of proof sketch or motivating idea, and articles about theorems famous for their multiple proofs should describe or include the most noteworthy ones (to the extent

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I don't think making readers skim past roughly the current quantity of text about various proofs is necessarily a problem – the proofs are important and insightful – but we should make some effort to make reading through the text pleasant and comprehensible. More important in my opinion is to find

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This page doesn't need more proofs, unless they are (a) published in reliable sources, and (b) in some way particularly notable or interesting, as described in reliable sources. We already have more than enough proofs to make the general point that the possible list of proofs is endless. With that

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I don't know how to minimize the infobox, and I'm using the default theme. Could you try using the DevTools to decrease the view width to see if that breaks it (you need to reload the page after you change the view)? I've tested it on my Android phone and on macOS, both with Chrome, and I have the

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In your proof, you assume that sin α = cos β and sin γ = sin(α + β)=1 - I'm not sure that this assumptions are independent of Pythagorean theorem - you also didn't explain where you got these assumptions from? (from geometry - triangle?). Zimba assumptions was weaker than your - he use arbitrary x

2140:. They used a pure (mostly) trigonometric proof, using what they call a "waffle cone" geometric construction to arrive at the equation a + b = 2ab / sin (2a) = c. It would be nice to add this to the article, in the "Trigonometric Proofs" section. (I'm not sure how to present this proof myself.) — 1438:

The fact that a proof is sourced from a unreliable source does not means that there are not reliable sources for this proof. In fact, the Cut-the-knot page for the algebraic proof refers to several older sources (one is almost 2,000 years old). On the other hand, the Cut-the-knot page for Zimba's

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is standard from centuries on, and is independent from Pythagorean theorem. So, Zimba's definition has nothing new. As Pythagorean theorem is about right triangles, it is impossible to provide a proof that does not involve any right triangle. The trigonometric proof given in the article does not

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I'm inclined against this on somewhat different grounds: having an article specifically devoted to collecting proofs of the theorem seems likely to grow into a huge indiscriminate collection of proofs, something that I do not think would make for a good encyclopedia article. It would be a cruft

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I don't think a split is necessary. There's not so much material here that it can't fit in a single article, and proofs are obviously one of the main things to discuss about a theorem. The sections on proofs should definitely be better organized for narrative flow. This kind of list that slowly

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The sections on proofs are presently in the middle of the explanations of the theorem, its consequences and its applications. Readers interested in these aspects of the theorem have thus to skip a wall of text that can be interesting in the whole for very few readers only. So, an immediate very

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Until the middle of the 19th century, all axioms of mathematics were abstractions of empirical experiments. You are talking of the relationship between the parallel postulate and the Pythagorean theorem. It is true that for proving the Pythagorean theorem, one needs the parallel postulate or

1162:. I'm not sure why he couldn't have simply specified the value of those functions at those points and then shown that the subtraction formulas still work when one or more of their inputs are in this expanded domain. Perhaps he considered that less elegant than the approach he did take. 1743:, but otherwise it works. However, big picture, I am neutral as to whether this is sufficiently noteworthy and interesting; if no other editor chimes in, don't let me stop you from deleting the section. (But if there is some support, maybe let's mend it rather than end it.) Thanks — 2874:

If the alternative way is keeping them in the article, the scenario I imagined would probably restructure sections in which the article presents the statement of theorem and its converse firstly and then a single proof of the theorem, and add the link, redirecting the latter section.

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The criterion for it is that is very short, simple and use only calculations without involving geometry (in direct way) like other proofs. So it can be very useful especially for people who hat not goot geometrical intuition (so we are dealing here with usability for a wider

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demonstrates a trigonometric proof of the Pythagorean theorem recently discovered by Calcea Johnson and Ne'Kiya Jackson, two high school students at St. Mary's Academy in New Orleans, who recently presented it at the (2023?) Spring Southeastern Sectional Meeting of the

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I am curious. Does Zimba claim to be the first to observe that the angle-subtraction formulas for sine and cosine can be proved without assuming the Pythagorean theorem? Does Zimba claim to be the first to observe that the subtraction formulas can be used to prove

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I think the grab-bag articles should generally be avoided where it's relatively straightforward to do so. They typically end up turning into substantially useless unreadable sludge. In the case where there is some important reference material involved, e.g.

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cannot be used because the trigonometric definition of sine as ration of opposite side to hypotenuse does not apply, namely, you cannot have two right angles inside a right triangle! Zimba was careful to note that trigonometric functions of angles

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I agree in spirit that some proofs should remain behind, with a pointer to the (new) main article that has those and additional proofs. I might haggle over whether it should be 5–6 vs. 2–3 that survive in the present article, but that's just

2093:(2) There is a two-panel diagram here with an upper and a lower panel. But the text talks about the lower panel first, then the upper, which is confusing. The diagram should be cut in half and made into two, rearranged in the logical order. 2442:

In any event, it is certainly the case that you could reshuffle your set of axioms to include the Pythagorean relation, if you wanted to. I'm not sure to what extent, if any, discussing this point is super useful in the context this page.

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If you want this perspective to be represented in the article, you are going to need to find published and scholarly sources that express the same sentiments. We cannot add material based purely on the musings of random Knowledge editors.

2848:, some readers might be willing to wade through that to find a point they are looking for (though I question how many), but for something like a list of proofs this doesn't seem that valuable to me. I would instead just direct readers to 1420:

In both proofs in this source there is information about who is considered to be the first author of the proof (12th century Hindu mathematician Bhaskara, and Jason Zimba) - although in both proofs on Knowledge this information is not

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should only include proofs with significant historical recognition, not recent flash-in-the-pan media hype and even more not primary sourced but otherwise non-notable proofs vaguely connected to recent flash-in-the-pan media hype. —

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Yes, the new article could become a grab-bag, but I think that that is okay. If the user has come looking for proofs, let's give them proofs. We'll have some minimum standards of course, but we can make the threshold a

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I tried making such a change. We can discuss whether it's worth it to make article content compromises for this, or if there's some other work around, and possibly revert that change. Does that at least fix the problem?

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I don't understand either. When I view the article on a mobile device (using the Android app on my phone) I do see the formula where it should be, immediately below the first paragraph and above the (minimized) infobox.

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Given that the norm of a cross product is a sine times the vector lengths and the dot product is a cosine times the vector lengths, it is pretty straightforward to plug these into a Pythagorean theorem. I'd do it with

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This is missing the point. Arguments here for why it's a good proof are not what is needed to justify its inclusion. If nobody has written secondary sources singling it out as a good proof, we cannot include it.

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I think that if your sin/cos funtions are the same as Zimba sin/cos functions (at least in (0,pi/2)) then whe shoud not refer to they definitions when we compare proofs - because you both uses same functions.

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and y angles and assume only that 0 < y < x < pi/2. (so he did not have to refer to any geometrical figure). This is why Zimba proof is quite interesting and qualitative different from other proofs.

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I remember that the list of all proofs may be suggested to relocate them into the WikiBooks. If this is a good idea, maybe we can add the link in the external link. However, I prefer to hear from others.

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But back to the proofs themselves - his proof is just pure symbolic and base only on sin/cos properties (substraction formulas) (which is somehow beautiful), your proof (I supose) need to relate to some

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a secondary source that attests that this is really new. This is really unlikely that this will ever occur for the following reason. The fundational principle on which is based trigonometry is that the

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Proofs aren't particularly helpful for validating most statement in most encyclopedia articles. In articles about a broad topic or field of study it's sometimes worth having a short proof or two

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The claims of being especially simple or of being the first non-circular trigonometric proof need secondary sources. We cannot make those claims based only on the original primary publication. —

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I suggest to remove this section. I have never heard of a relationship between Pythagorean theorem and the cross product, and I do not see in the section any indication of such a relationship.

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clear sources for every proof, ideally mention who first made each proof and link to the original, make the formatting and illustrations a bit more orderly and maybe more consistent in style.

321: 3040:), and I also believe that some proofs of the Pythagorean theorem do too, but certainly not all (or at least not in this article). However this theorem is unique in that there have been so 1450:

and similarity of right triangles. The latter is simple and direct, while Zimba's proof requires an elaborated geometrical construction and the proof of an auxiliary trigonometric formula.

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The information about "first non-circular trigonometric proof" was not included into deleted proof (in the same way like "primality" (in some way) of the some other proofs on this page).

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said though, this is a fine proof. Nice work. If you can find some website that attempts to comprehensively list as many proofs as possible, you could submit this there. (Unfortunately

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It seems that the main reason for a split is that, without a split, much more work is needed to reach the good-article status. I do not know whether tis is a good reason for a split

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Please, move the first formula on the page to go right after the first paragraph (if you view the page on a mobile device now, you will not see the formula where it should be).

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Simplicity is obvious because tricky part is only adding zero by: x-(x-y) (and use some old known formulas) - I doubt anyone will describe such obvious things in an article.

1999: 1178:? Does Zimba claim to be the first to put these two thoughts together? Does Zimba claim that his approach is distinct from previous approaches because he avoided using 2787:

accretes inconsistent items without curation is pretty common among popular older pages. I just tried to do some cleanup on the somewhat similar list of derivations at

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should focus on explaining the Pythagorean theorem. However, at this point, the article also contains lots of proof of this theorem. In that case, should both sections

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I'd say the "idea of using axioms that were empirically motivatable" was most of the spirit of Euclid's axioms (and various alternatives over the following centuries).

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in your proof you use a,b,c (from geometry object - triangle) - but Zimba use only two arbitrary angles x and y (without involving geometry in direct way like you).

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points out above we need reliable sources for every proof we publish, and it seems to me that rigid enforcement of this would deal with the "cruft" problem.

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Well, Zimba's proof has also been included in this source which you found reliable (because you allowed this source to be used on this page for many years)

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Trigonometry is not inherently based on the Pythagorean theorem. Much of it is, but nowhere near the entirety. After all, the field preceded Pythagoras (

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claims that the Pythagorean theorem is equivalent to the fifth postulate. I've added a brief blurb to this effect in the Pythagorean theorem article. —

311: 2217: 1558:– To be precise, this definition dates from about the middle of the 18th century, and became standard somewhere around the middle of the 19th century. – 2337:

to force the equation to stay with the paragraph. We could perhaps try adding a paragraph break earlier so that the sentence stays with the equation. –

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Regarding "his proof is just pure symbolic ... your proof (I supose) need to relate to some triangle." He uses triangles, but I suppose that you mean

3303: 3008:. As others here will know, there has been much discussion about when and whether proofs should be included in mathematics articles (see for example: 1521:

I'm surprised by what you write - can you provide a link (or explain it) to a trigonometric proof which not require any other trigonometric identity?

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Therefore, it can be consistently assumed that information about Zimba's proof is based on reliable sources (unless you have double standards)

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Yes, we'd need a secondary source to make any claim that a proof was 'first'. We can't rely on what editors happen to have found themselves.

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Zimba only shows that functions sin/cos can be defined independent of Pythagorean theorem, to be sure that using them in proofs is allowed.

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depend only on one acute angle of a right triangle, and do not depend on the size of the riangle. This is directly used in the proofs of

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These are technical reason for not including Zimba's proof, but, again, the main reason for not including it is that inclusion requires

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This is fairly simple, so I like that. I see that the triangles are similar, but we'd want to explain that. I do hope you find a

2322::", and instead of the the equation you will see this box with information about the theorem. The formula will be below that box. 3298: 3288: 3109:

Wikibook projects for proofs or set up a dedicated Wikibook project just for this collection. As an external option there is the

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something equivalent. But the converse is not true, since the Pythagorean requires a notion of distance. In particular, in an

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I'm not sure that Zimba was first - but if not, then should exists similar results before him. But so far I haven't found any

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That's in conjunction with postulate 3 saying you can draw circles (and various other assumptions left unstated by the

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https://en.wikipedia.org/search/?title=Pythagorean_theorem&oldid=1149322678#Jason_Zimba_trigonometric_proof%5B25%5D

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the article is with good support from secondary sourcing - and not in the form of self-published arxiv stuff. -

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proofs discovered (or created ;-)), so that, to me, an article devoted to them seems warranted. Of course, as

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useful action would be to move these sections toward the end of the article, possibly with a link in the lead.

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I understand your "reverse perspective" as the study of the axioms that are needed for proving some theorems.

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Knowledge:Redirects for discussion/Log/2023 April 29 § Pythagoras' theorem proof (rational trigonometry)

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list, and, often, the headings do not give the needed information on the specifity of the proof method.

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The sections on proofs require to be restructured and largely rewritten. Presently they appear as an

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without referring to a right triangle, though he needs a right triangle for the next step, to get to

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on Knowledge. If you would like to participate, please visit the project page, where you can join

3227: 3209: 3197: 2777: 2732: 2492: 2482: 2422: 2197: 1721: 1544: 1512: 1489: 827:, the shorter-than-Zimba proof gets even shorter. With the angle-addition formula for sines and 792: 536:

In the other side, for historical point of view, this is also first known trigonometrical proof.

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gives true math, but it isn't closely enough related to the Pythagorean theorem. Specifically,

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harder to push back against adding this or that arbitrary proof that anyone wants to include. –

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The Zimba proof relies on the angle-addition formula for sines. However with that formula and

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That is not reliably published. And it has no depth in its coverage of the Zimba publication. —

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triangles, we can write the following products and ratios relationships, then multiply them:

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of geometry, although he chose to use similar triangles rather than the Pythagorean theorem.

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project. Our article should offer links to such collections in the external links section.--

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readability-improvement work on the material currently in the article seems more important.

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I have no clear opinion whether the article must be split. However, here are some comments.

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Well, the idea of using axioms that were empirically motivatable was part of the spirit of

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Trigonometry is based on Pythagorean theorem. Therefore, a trigonometric proof should be

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There are at least a couple relevant relationships. First, for any two Euclidean vectors

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Also, the last sentence of Zimba's introduction suggest that his aim is to prove the

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There are also some trigometric proofs of the theorem. These could be mentioned.

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Try opening the page on a phone. The first paragraph end with "...often called the

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You use sin, cos and γ, α, β with asumption sin α = cos β and sin γ = sin(α + β)=1

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Ok, here is secondary source which mention that this is first trigonometric proof:

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Thanks for the great feedback so far, will look into making these improvements.

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I agree with David Eppstein; the Zimba proof is insufficiently noteworthy. —

2464:, the parallel postulate is verified, but there is no notion of right angle. 3200:. Nevertheless, if you know a trigonometric proof that is not circular and 3069:

practical; obviously some proofs are extremely long or technical). Clearly

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Yes, now the formula is after the paragraph, which is after the infobox.

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He use sin, cos and angles x,y with asumption 0 < y < x < pi/2.

1469:. This suggests that his article is not primarily about a proof of the 1403: 600: 2522:, such as that a circle intersects every line through its center). – 599:"OTHER TRIGONOMETRIC PROOFS ON PYTHAGORAS THEOREM", N. Luzia, 2015, 2085:

Algebraic Proofs: edit request to number and rearrange the diagram

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Why short Zimba trigonometric proof (main idea) from this revision

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lower than it is for proofs that are presently in this article. —

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Relatedly, if you start with two vectors which are perpendicular

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require subtraction formula or any other trigonometric identity.

1484:, and Zimba's article is not notable enough for being mentioned. 1082:" in his notation) is from a right-triangle when he argues that 3253:

There has been some previous discussion about this topic. See

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Talk:Pythagorean theorem/Archive 7 § Proof using trigonometry

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rather than as validation for claims made. But in an article

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follow immediately from the definitions that Zimba gives for

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of this article to be created. For further information, see

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Create an article for proof of Pythagorean theorem's only

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as he defines them are defined only on the open interval

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Knowledge talk:WikiProject Mathematics/Proofs/Archive 1

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Knowledge talk:WikiProject Mathematics/Proofs/Archive 1

2188: 1632:, doesn't look anything like the Pythagorean theorem's 1477:

can be easily modified for proving both simultaneously.

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Dividing them naturally also gives us: a' + b' = c'

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is a rather complete study of this kind of questions.

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Pythagorean theorem#Proofs using constructed squares

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I don't understand the request. Can you elaborate? –

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to determine whether its use and function meets the

282:, a collaborative effort to improve the coverage of 1593:

Pythagorean theorem § Relation to the cross product

1537:§ Trigonometric proof using Einstein's construction 1502:The definition of trigonometric functions given in 1475:§ Trigonometric proof using Einstein's construction 1444:§ Trigonometric proof using Einstein's construction 785:§ Trigonometric proof using Einstein's construction 2610: 2551:Looking at the hypotenuse and height of the three 2061: 1993: 1956: 1855: 1801: 1778: 999: 972: 951: 367:The rationale behind the request is: "Important". 3258: 2189:Pythagoras' theorem proof (rational trigonometry) 2177:Pythagoras' theorem proof (rational trigonometry) 2155:See multiple long discussions above, starting at 1642:We could improve this by changing occurrences of 2559:a·a' + b·b' = c·c' (products = 2 x areas) 1308:triangles. Yes, agreed, he gets all the way to 3319:Knowledge level-4 vital articles in Mathematics 2725:Pythagorean theorem#Other proofs of the theorem 661:, the result is more immediate: one can insert 2540:Simple algebraic proof using similar triangles 1268:) not contains information that it was first. 71:If it no longer meets these criteria, you can 3028:. But some do (e.g. the irrationality of the 3010:Knowledge talk:WikiProject Mathematics/Proofs 2156: 1607:are perpendicular to each other the value of 1410:- so you consider this source to be reliable. 447:This page has archives. Sections older than 175:Former featured article, current good article 8: 3136:reliable collection in Knowledge itself in 1442:Also, comparing Zimba's proof with that of 3026:most proofs have little encyclopedic value 841:, the result is immediate: one can insert 228: 85: 15: 2602: 2589: 2576: 2570: 2223:Semi-protected edit request on 7 May 2023 2050: 2037: 2024: 2006: 1971: 1945: 1940: 1925: 1916: 1911: 1896: 1887: 1882: 1870: 1868: 1818: 1791: 1771: 987: 985: 965: 932: 3309:Knowledge vital articles in Mathematics 2648:died a few years ago, so I don't think 1439:article refers only to Zimba's article. 457:when more than 10 sections are present. 230: 3324:GA-Class vital articles in Mathematics 2062:{\displaystyle (a+b)^{2}=a^{2}+b^{2}.} 1856:{\displaystyle ab=a\wedge b+a\cdot b,} 1555: 2901:too long and too hard to navigate. — 2668:that shows that this is sufficiently 7: 601:https://arxiv.org/pdf/1502.06628.pdf 354:, submitted by Lionsdude148, for an 276:This article is within the scope of 191: 189: 2731:? The fact I have discussed in the 1399:I found a solution to this impasse. 3284:Knowledge former featured articles 2562:a/a' = b/b' = c/c' (ratios) 1467:Pythagorean trigonometric identity 1455:Pythagorean trigonometric identity 1406:section there is a proof based on 14: 3334:Top-priority mathematics articles 3138:Proofs of the Pythagorean theorem 2929:Proofs of the Pythagorean theorem 2927:The new article's title might be 2611:{\displaystyle a^{2}+b^{2}=c^{2}} 1128:, Zimba uses an unrelated angle " 1000:{\displaystyle {\frac {\pi }{2}}} 451:may be automatically archived by 296:Knowledge:WikiProject Mathematics 64:. If you can improve it further, 3304:Knowledge level-4 vital articles 2846:list of trigonometric identities 2230: 1402:Currently in the article in the 381: 341: 299:Template:WikiProject Mathematics 263: 253: 232: 199: 190: 42: 19: 3314:GA-Class level-4 vital articles 3038:Gödel's incompleteness theorems 2386:Reverse mathematics perspective 781:§ Proof using similar triangles 316:This article has been rated as 3259:§ Why Zimba proof was deleted? 2090:as much when edits are done. 2021: 2008: 1941: 1926: 1912: 1897: 1883: 1871: 952:{\displaystyle \sin \gamma =1} 219:It is of interest to multiple 52:has been listed as one of the 1: 3329:GA-Class mathematics articles 2729:Proofs of Pythagorean theorem 2652:is taking new submissions.) – 2206:until a consensus is reached. 2138:American Mathematical Society 1650:. In that case the equation 1587:Relation to the cross product 1135:I see that Zimba argues that 1036:As I see it, the opposite of 290:and see a list of open tasks. 2650:cut-the-knot.org/pythagoras/ 2530:21:05, 12 January 2024 (UTC) 2514:20:49, 12 January 2024 (UTC) 2487:20:39, 12 January 2024 (UTC) 2451:21:02, 12 January 2024 (UTC) 2435:18:57, 12 January 2024 (UTC) 1599:Because the sides of length 1114:. (In contrast, instead of 464:Why Zimba proof was deleted? 361:WikiProject Spoken Knowledge 3202:has been reliably published 2855:The Pythagorean Proposition 2850:cut-the-knot.org/pythagoras 2417:21:07, 8 January 2024 (UTC) 2401:19:13, 8 January 2024 (UTC) 2253:to reactivate your request. 2241:has been answered. Set the 1994:{\displaystyle a\cdot b=0,} 1556:standard from centuries on, 3355: 3174:05:26, 16 March 2024 (UTC) 3159:14:32, 14 March 2024 (UTC) 3123:23:08, 13 March 2024 (UTC) 3103:13:28, 14 April 2024 (UTC) 3081:15:04, 13 March 2024 (UTC) 3056:13:43, 13 March 2024 (UTC) 3034:Cantor's diagonal argument 2995:12:36, 13 March 2024 (UTC) 2980:19:41, 12 March 2024 (UTC) 2965:18:49, 12 March 2024 (UTC) 2950:16:33, 12 March 2024 (UTC) 2920:15:26, 12 March 2024 (UTC) 2885:12:27, 13 March 2024 (UTC) 2870:16:35, 12 March 2024 (UTC) 2839:16:27, 12 March 2024 (UTC) 2807:14:39, 12 March 2024 (UTC) 2782:13:12, 12 March 2024 (UTC) 2749:10:20, 12 March 2024 (UTC) 2727:be split into the article 2169:07:16, 24 April 2023 (UTC) 2157:§ Proof using trigonometry 2150:22:57, 23 April 2023 (UTC) 2117:19:46, 14 April 2023 (UTC) 2109:lettered as an outline". – 2103:14:20, 14 April 2023 (UTC) 2077:18:31, 14 April 2023 (UTC) 1762:13:41, 14 April 2023 (UTC) 1726:17:12, 13 April 2023 (UTC) 1711:16:38, 13 April 2023 (UTC) 1566:18:46, 14 April 2023 (UTC) 1549:10:05, 15 April 2023 (UTC) 1531:18:35, 14 April 2023 (UTC) 1517:18:20, 14 April 2023 (UTC) 1494:10:02, 15 April 2023 (UTC) 1461:, rather that proving the 1434:08:14, 15 April 2023 (UTC) 1381:18:37, 14 April 2023 (UTC) 1367:18:15, 14 April 2023 (UTC) 1293:18:06, 14 April 2023 (UTC) 1278:18:02, 14 April 2023 (UTC) 1260:17:53, 14 April 2023 (UTC) 1246:17:32, 14 April 2023 (UTC) 1216:16:22, 14 April 2023 (UTC) 1032:14:48, 14 April 2023 (UTC) 1017:11:36, 15 April 2023 (UTC) 924:13:35, 14 April 2023 (UTC) 811:09:49, 14 April 2023 (UTC) 797:21:39, 12 April 2023 (UTC) 769:20:26, 12 April 2023 (UTC) 628:20:54, 11 April 2023 (UTC) 612:20:37, 11 April 2023 (UTC) 592:20:26, 11 April 2023 (UTC) 577:20:19, 11 April 2023 (UTC) 560:20:06, 11 April 2023 (UTC) 546:20:02, 11 April 2023 (UTC) 528:19:49, 11 April 2023 (UTC) 501:19:39, 11 April 2023 (UTC) 485:19:14, 11 April 2023 (UTC) 110:Refreshing brilliant prose 3339:Spoken Knowledge requests 3294:Mathematics good articles 2705:21:21, 7 March 2024 (UTC) 2691:17:41, 7 March 2024 (UTC) 2660:16:57, 7 March 2024 (UTC) 2638:16:20, 7 March 2024 (UTC) 1007:cannot be directly used. 315: 248: 227: 172: 163:Good article reassessment 88: 84: 56:Mathematics good articles 37:) and why it was removed. 3062:as illustrative examples 2194:redirects for discussion 2181:Redirects for discussion 1863:and these parts satisfy 1264:Yep, but deleted proof ( 322:project's priority scale 3299:GA-Class vital articles 3289:Knowledge good articles 3269:16:11, 6 May 2024 (UTC) 3245:15:26, 9 May 2024 (UTC) 3232:15:21, 9 May 2024 (UTC) 3214:08:16, 6 May 2024 (UTC) 3191:01:03, 6 May 2024 (UTC) 2735:, and IMO this regards 2372:18:55, 7 May 2023 (UTC) 2358:18:50, 7 May 2023 (UTC) 2345:18:42, 7 May 2023 (UTC) 2332:18:31, 7 May 2023 (UTC) 2312:18:39, 7 May 2023 (UTC) 2297:18:30, 7 May 2023 (UTC) 2282:18:25, 7 May 2023 (UTC) 2269:18:11, 7 May 2023 (UTC) 2218:06:59, 7 May 2023 (UTC) 2123:New trigonometric proof 279:WikiProject Mathematics 125:Featured article review 31:former featured article 3132: 2715:Note that the article 2612: 2548: 2063: 1995: 1958: 1857: 1803: 1780: 1001: 974: 953: 454:Lowercase sigmabot III 35:the nomination archive 3204:, is could be added. 3128: 3090:per above discussion. 2613: 2547: 2064: 1996: 1959: 1858: 1804: 1781: 1620:The right-hand side, 1002: 975: 954: 206:level-4 vital article 62:good article criteria 2710: 2569: 2320:Pythagorean equation 2005: 1970: 1867: 1817: 1790: 1770: 1504:Trigonometric ratios 1448:Trigonometric ratios 984: 964: 931: 777:trigonometric ratios 302:mathematics articles 144:Good article nominee 2717:Pythagorean theorem 2646:Alexander Bogomolny 2198:redirect guidelines 2192:has been listed at 1471:Pythagorean theorem 1463:Pythagorean theorem 1459:Pythagorean theorem 1074:. Zimba uses that 1040:is the adjacent of 50:Pythagorean theorem 27:Pythagorean theorem 3182:Trigometric proofs 2608: 2549: 2493:parallel postulate 2059: 1991: 1954: 1853: 1802:{\displaystyle b,} 1799: 1776: 1457:without using the 997: 970: 949: 815:If you don't want 271:Mathematics portal 215:content assessment 89:Article milestones 2789:quadratic formula 2763:WP:indiscriminate 2474:Geometric algebra 2423:Birkhoff's axioms 2257: 2256: 1811:geometric product 1779:{\displaystyle a} 1523:Kamil Kielczewski 1426:Kamil Kielczewski 1373:Kamil Kielczewski 1300:Kamil Kielczewski 1270:Kamil Kielczewski 1238:Kamil Kielczewski 1024:Kamil Kielczewski 995: 973:{\displaystyle 0} 803:Kamil Kielczewski 604:Kamil Kielczewski 569:Kamil Kielczewski 538:Kamil Kielczewski 477:Kamil Kielczewski 461: 460: 373: 372: 336: 335: 332: 331: 328: 327: 184: 183: 180: 179: 80: 3346: 3146: 3030:Square root of 2 3022:this search list 2937: 2907: 2852:, Loomis (1968) 2826: 2678: 2617: 2615: 2614: 2609: 2607: 2606: 2594: 2593: 2581: 2580: 2501: 2491:The page on the 2364:Germanivanov0719 2324:Germanivanov0719 2304:Germanivanov0719 2261:Germanivanov0719 2248: 2244: 2234: 2233: 2227: 2216: 2211: 2191: 2068: 2066: 2065: 2060: 2055: 2054: 2042: 2041: 2029: 2028: 2000: 1998: 1997: 1992: 1963: 1961: 1960: 1955: 1950: 1949: 1944: 1929: 1921: 1920: 1915: 1900: 1892: 1891: 1886: 1874: 1862: 1860: 1859: 1854: 1808: 1806: 1805: 1800: 1785: 1783: 1782: 1777: 1749: 1742: 1736: 1698: 1691: 1677: 1649: 1645: 1637: 1631: 1616: 1606: 1602: 1404:Algebraic proofs 1354: 1347: 1339: 1319: 1303: 1203: 1196: 1189: 1185: 1181: 1177: 1161: 1154: 1150: 1142: 1138: 1131: 1127: 1113: 1093: 1081: 1077: 1073: 1069: 1065: 1054: 1043: 1039: 1006: 1004: 1003: 998: 996: 988: 979: 977: 976: 971: 958: 956: 955: 950: 911: 904: 893: 870: 862: 851: 840: 826: 822: 818: 756: 749: 729: 706: 698: 679: 660: 515: 456: 440: 385: 377: 345: 338: 304: 303: 300: 297: 294: 273: 268: 267: 257: 250: 249: 244: 236: 229: 212: 203: 202: 195: 194: 193: 186: 173:Current status: 158: 139: 137:December 9, 2005 105:January 19, 2004 86: 69: 46: 23: 16: 3354: 3353: 3349: 3348: 3347: 3345: 3344: 3343: 3274: 3273: 3184: 3142: 3066:about a theorem 2933: 2903: 2822: 2713: 2674: 2666:reliable source 2619: 2598: 2585: 2572: 2567: 2566: 2542: 2497: 2388: 2246: 2242: 2231: 2225: 2212: 2207: 2187: 2184: 2125: 2087: 2046: 2033: 2020: 2003: 2002: 1968: 1967: 1939: 1910: 1881: 1865: 1864: 1815: 1814: 1788: 1787: 1768: 1767: 1745: 1738: 1732: 1694: 1679: 1651: 1647: 1643: 1633: 1621: 1617:is always zero. 1608: 1604: 1600: 1589: 1473:. In any case, 1350: 1341: 1321: 1309: 1297: 1199: 1191: 1187: 1183: 1179: 1167: 1156: 1152: 1144: 1140: 1136: 1129: 1115: 1095: 1083: 1079: 1075: 1071: 1067: 1056: 1045: 1041: 1037: 982: 981: 962: 961: 929: 928: 907: 895: 872: 864: 853: 842: 828: 824: 820: 816: 752: 731: 708: 700: 681: 662: 648: 511: 466: 452: 441: 435: 390: 369: 301: 298: 295: 292: 291: 269: 262: 242: 213:on Knowledge's 210: 200: 156:October 6, 2007 154: 135: 12: 11: 5: 3352: 3350: 3342: 3341: 3336: 3331: 3326: 3321: 3316: 3311: 3306: 3301: 3296: 3291: 3286: 3276: 3275: 3272: 3271: 3251: 3250: 3249: 3248: 3247: 3183: 3180: 3179: 3178: 3177: 3176: 3133: 3106: 3105: 3093: 3092: 3091: 3084: 3083: 3002: 3001: 3000: 2999: 2998: 2997: 2972:David Eppstein 2952: 2922: 2898:Support split: 2895: 2894: 2893: 2892: 2891: 2890: 2889: 2888: 2887: 2860:alternate scan 2813: 2796: 2792: 2771: 2770: 2769: 2766: 2759: 2712: 2709: 2708: 2707: 2693: 2662: 2628: 2625: 2621: 2605: 2601: 2597: 2592: 2588: 2584: 2579: 2575: 2564: 2558: 2541: 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1400: 1389: 1388: 1387: 1386: 1385: 1384: 1383: 1234: 1230: 1227: 1224: 1221: 1163: 1133: 1019: 1009:Danko Georgiev 994: 991: 969: 948: 945: 942: 939: 936: 799: 645: 644: 643: 642: 641: 640: 639: 638: 637: 636: 635: 634: 633: 632: 631: 630: 620:David Eppstein 597: 584:David Eppstein 565: 552:David Eppstein 534: 530: 504: 503: 493:David Eppstein 474:David Eppstein 472:was deleted? @ 465: 462: 459: 458: 446: 443: 442: 437: 433: 431: 428: 427: 392: 391: 386: 380: 371: 370: 348: 346: 334: 333: 330: 329: 326: 325: 314: 308: 307: 305: 288:the discussion 275: 274: 258: 246: 245: 237: 225: 224: 218: 196: 182: 181: 178: 177: 170: 169: 166: 159: 151: 150: 147: 140: 132: 131: 128: 121: 120:March 20, 2004 117: 116: 113: 106: 102: 101: 98: 95: 91: 90: 82: 81: 47: 39: 38: 24: 13: 10: 9: 6: 4: 3: 2: 3351: 3340: 3337: 3335: 3332: 3330: 3327: 3325: 3322: 3320: 3317: 3315: 3312: 3310: 3307: 3305: 3302: 3300: 3297: 3295: 3292: 3290: 3287: 3285: 3282: 3281: 3279: 3270: 3267: 3264: 3260: 3256: 3252: 3246: 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1657: 1653: 1641: 1634: 1627: 1623: 1613: 1609: 1591:The section 1590: 1582: 1351: 1335: 1331: 1327: 1323: 1315: 1311: 1305: 1200: 1192: 1173: 1169: 1157: 1146: 1124: 1120: 1116: 1109: 1105: 1101: 1097: 1089: 1085: 1062: 1058: 1051: 1047: 908: 901: 897: 890: 886: 882: 878: 874: 866: 859: 855: 848: 844: 837: 833: 829: 753: 745: 741: 737: 733: 726: 722: 718: 714: 710: 702: 695: 691: 687: 683: 676: 672: 668: 664: 657: 653: 649: 646: 512: 467: 448: 395: 387: 374: 366: 360: 349: 318:Top-priority 317: 277: 243:Top‑priority 221:WikiProjects 204: 174: 162: 161: 143: 142: 123: 108: 72: 70: 66:please do so 54: 53: 49: 26: 3096:Youprayteas 3050:Paul August 1737:instead of 1678:, would be 1408:this source 1371:yep, agree 350:There is a 293:Mathematics 284:mathematics 240:Mathematics 3278:Categories 3164:editors.-- 2987:Dedhert.Jr 2957:XOR'easter 2877:Dedhert.Jr 2741:Dedhert.Jr 2469:Emil Artin 2243:|answered= 2142:Loadmaster 1348:Agreed! — 60:under the 3263:jacobolus 3239:jacobolus 3111:ProofWiki 3075:jacobolus 3046:jacobolus 2864:jacobolus 2862:), etc. – 2801:jacobolus 2697:Weallwiki 2654:jacobolus 2630:Weallwiki 2524:jacobolus 2445:jacobolus 2352:jacobolus 2339:jacobolus 2276:jacobolus 2111:jacobolus 2095:editeur24 2071:jacobolus 1560:jacobolus 1421:provided. 1233:triangle. 1094:leads to 533:audience) 209:is rated 3224:Dan Wang 3206:D.Lazard 3198:circular 3155:contribs 3147:uantling 2946:contribs 2938:uantling 2925:Comment: 2916:contribs 2908:uantling 2835:contribs 2827:uantling 2812:details. 2774:D.Lazard 2687:contribs 2679:uantling 2520:Elements 2510:contribs 2502:uantling 2479:D.Lazard 2471:'s book 1758:contribs 1750:uantling 1718:D.Lazard 1707:contribs 1699:uantling 1541:D.Lazard 1535:Look at 1509:D.Lazard 1486:D.Lazard 1393:D.Lazard 1363:contribs 1355:uantling 1212:contribs 1204:uantling 1078:(well, " 920:contribs 912:uantling 896:1 = sin 894:to give 789:D.Lazard 765:contribs 757:uantling 730:to give 524:contribs 516:uantling 449:365 days 388:Archives 211:GA-class 74:reassess 3188:Bubba73 2670:notable 2553:similar 1397:MrOllie 1344:MrOllie 1285:MrOllie 1252:MrOllie 352:request 320:on the 130:Demoted 97:Process 3166:Kmhkmh 3115:Kmhkmh 2818:little 2737:GACR2a 2733:WT:GAN 1668:‖ = (‖ 1338:) = 1 1314:+ cos 1172:+ cos 1155:or at 1088:+ cos 1061:= sin 1050:= cos 900:+ cos 885:+ sin 877:= sin 863:, and 858:= cos 847:= sin 721:+ sin 713:= sin 699:, and 686:= sin 667:= cos 217:scale. 149:Listed 100:Result 3220:seked 2247:|ans= 2237:This 2129:video 2127:This 1660:) + ‖ 1330:) + ( 1306:right 1123:/2 − 1104:) + ( 871:into 740:) + ( 732:1 = ( 707:into 397:Index 198:This 29:is a 3257:and 3228:talk 3210:talk 3170:talk 3151:talk 3140:. — 3119:talk 3042:many 2991:talk 2976:talk 2961:talk 2942:talk 2931:. — 2912:talk 2881:talk 2831:talk 2778:talk 2745:talk 2723:and 2701:talk 2683:talk 2634:talk 2506:talk 2483:talk 2431:talk 2413:talk 2397:talk 2368:talk 2328:talk 2308:talk 2293:talk 2265:talk 2165:talk 2146:talk 2099:talk 1809:the 1786:and 1754:talk 1722:talk 1703:talk 1689:(ac) 1685:(ab) 1681:(aa) 1603:and 1545:talk 1527:talk 1513:talk 1490:talk 1430:talk 1415:here 1377:talk 1359:talk 1310:sin 1289:talk 1274:talk 1266:here 1256:talk 1242:talk 1208:talk 1197:? — 1190:and 1182:and 1168:sin 1145:(0, 1139:and 1084:sin 1070:and 1057:cos 1055:and 1046:sin 1028:talk 1013:talk 916:talk 889:cos 881:cos 873:sin 865:sin 854:sin 843:cos 823:and 807:talk 793:talk 783:and 761:talk 750:. — 725:cos 717:cos 709:sin 701:sin 682:cos 663:sin 624:talk 608:talk 588:talk 573:talk 556:talk 542:talk 520:talk 497:talk 481:talk 168:Kept 115:Kept 94:Date 3266:(t) 3261:. – 3242:(t) 3222:). 3078:(t) 3071:all 2867:(t) 2804:(t) 2672:. — 2657:(t) 2527:(t) 2448:(t) 2355:(t) 2342:(t) 2279:(t) 2245:or 2209:Jay 2131:by 2114:(t) 2074:(t) 1813:is 1672:‖ ‖ 1646:to 1626:‖ ‖ 1563:(t) 1340:. 1318:= 1 1186:at 1184:cos 1180:sin 1176:= 1 1149:/2) 1141:cos 1137:sin 1132:".) 1092:= 1 1072:cos 1068:sin 980:or 935:sin 905:. — 869:= 1 705:= 1 312:Top 3280:: 3230:) 3212:) 3172:) 3157:) 3153:| 3121:) 3036:, 3032:, 3016:, 3012:, 2993:) 2978:) 2963:) 2948:) 2944:| 2918:) 2914:| 2883:) 2837:) 2833:| 2780:) 2747:) 2703:) 2689:) 2685:| 2636:) 2512:) 2508:| 2485:) 2433:) 2415:) 2399:) 2370:) 2330:) 2310:) 2295:) 2267:) 2251:no 2214:💬 2167:) 2148:) 2101:) 1977:⋅ 1934:⋅ 1905:∧ 1845:⋅ 1833:∧ 1760:) 1756:| 1724:) 1709:) 1705:| 1687:= 1683:+ 1676:‖) 1664:× 1656:· 1612:· 1547:) 1539:. 1529:) 1515:) 1492:) 1432:) 1379:) 1365:) 1361:| 1291:) 1276:) 1258:) 1244:) 1214:) 1210:| 1195:/2 1160:/2 1119:= 1030:) 1015:) 990:π 941:γ 938:⁡ 922:) 918:| 852:, 836:+ 832:= 819:, 809:) 795:) 767:) 763:| 690:= 680:, 671:= 656:+ 652:= 626:) 610:) 590:) 575:) 558:) 544:) 526:) 522:| 499:) 483:) 422:, 418:, 414:, 410:, 406:, 402:, 364:. 77:it 68:. 3226:( 3208:( 3168:( 3149:( 3144:Q 3117:( 3053:☎ 2989:( 2974:( 2959:( 2940:( 2935:Q 2910:( 2905:Q 2879:( 2858:( 2829:( 2824:Q 2791:. 2776:( 2743:( 2699:( 2681:( 2676:Q 2632:( 2604:2 2600:c 2596:= 2591:2 2587:b 2583:+ 2578:2 2574:a 2504:( 2499:Q 2481:( 2443:– 2429:( 2411:( 2407:— 2395:( 2366:( 2350:– 2326:( 2306:( 2291:( 2287:— 2263:( 2175:" 2163:( 2159:— 2144:( 2097:( 2069:– 2057:. 2052:2 2048:b 2044:+ 2039:2 2035:a 2031:= 2026:2 2022:) 2018:b 2015:+ 2012:a 2009:( 1989:, 1986:0 1983:= 1980:b 1974:a 1952:. 1947:2 1942:| 1937:b 1931:a 1927:| 1923:+ 1918:2 1913:| 1908:b 1902:a 1898:| 1894:= 1889:2 1884:| 1879:b 1876:a 1872:| 1851:, 1848:b 1842:a 1839:+ 1836:b 1830:a 1827:= 1824:b 1821:a 1797:, 1794:b 1774:a 1752:( 1747:Q 1740:b 1734:c 1720:( 1701:( 1696:Q 1674:c 1670:a 1666:c 1662:a 1658:c 1654:a 1652:( 1648:c 1644:b 1638:. 1635:c 1630:‖ 1628:b 1624:a 1622:‖ 1614:b 1610:a 1605:b 1601:a 1543:( 1525:( 1511:( 1488:( 1428:( 1417:. 1395:@ 1391:@ 1375:( 1357:( 1352:Q 1346:: 1342:@ 1336:c 1334:/ 1332:b 1328:c 1326:/ 1324:a 1322:( 1316:x 1312:x 1302:: 1298:@ 1287:( 1272:( 1254:( 1240:( 1206:( 1201:Q 1193:π 1188:0 1174:α 1170:α 1158:π 1153:0 1147:π 1130:y 1125:α 1121:π 1117:β 1112:) 1110:c 1108:/ 1106:b 1102:c 1100:/ 1098:a 1096:( 1090:α 1086:α 1080:x 1076:α 1063:β 1059:α 1052:β 1048:α 1042:β 1038:α 1026:( 1011:( 993:2 968:0 947:1 944:= 914:( 909:Q 902:α 898:α 891:α 887:β 883:β 879:α 875:γ 867:γ 860:α 856:β 849:α 845:β 838:β 834:α 830:γ 825:c 821:b 817:a 805:( 791:( 759:( 754:Q 748:) 746:c 744:/ 742:b 738:c 736:/ 734:a 727:α 723:β 719:β 715:α 711:γ 703:γ 696:c 694:/ 692:b 688:β 684:α 677:c 675:/ 673:a 669:β 665:α 658:β 654:α 650:γ 622:( 606:( 586:( 582:— 571:( 554:( 540:( 518:( 513:Q 495:( 479:( 424:7 420:6 416:5 412:4 408:3 404:2 400:1 324:. 223:. 79:.

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