Talk:Exact trigonometric values

Neologism ?
The term "trigonometric number" appears to be a neologism. The result mentioned in the article is a version of Niven's theorem, but the only source I can find that calls these irrational numbers "trigonometric numbers" is this page at everthing2, which I don't think is a reliabe source. I have added a prod template to the article. Gandalf61 (talk) 10:24, 1 June 2008 (UTC)


 * I was going to respond that it's not a neologism since it's in Niven's well-known expository book. But I'm not finding it there.  I'm fairly sure I've seen it used in exactly this sense before, and the definition makes sense. Michael Hardy (talk) 14:48, 1 June 2008 (UTC)
 * OK, the book I was looking at was Irrational Numbers. I'm going to try that other one. Michael Hardy (talk) 15:08, 1 June 2008 (UTC)
 * Bingo. Chapter 5 of the book cited in the article is Trigonometric and Logarithmic Numbers.  And the sense in which the term is used is exactly that of this article. Michael Hardy (talk) 15:16, 1 June 2008 (UTC)
 * A term defined in one obscure book from the 1950s and never (or barely ever) used by anyone else again is a questionable thing for Wikipedia to canonize with an article. –jacobolus (t) 05:04, 25 November 2021 (UTC)
 * The concept of the real part of a root of unity is clearly notable. So the question is whether you can suggest a better title that is shorter than the red one. By the way this characterization of trigonometric numbers is important enough for the lead. I have just added it. D.Lazard (talk) 08:54, 25 November 2021 (UTC)

Delete / redirect?
I don't think this concept is really notable and independent enough to have its own page. It is just the real parts of the roots of unity. Googling for it turns up very little, and almost all the incoming wikilinks are because of the irrational number template. I suggest redirecting to Root_of_unity and putting a sentence there about the concept. (And removing it from the irrational number template.) Danstronger (talk) 22:03, 19 November 2021 (UTC)
 * I would suggest a merge rather than simply a redirect, because the target article deserves to be completed on this subject. Namely: the so-called trigonometric number are "solvable numbers" (that is, they can be expressed in radicals), the condition under which they are constructible must be stated, as well as the way of computing their minimal polynomial from the corresponding cyclotomic polynomial, etc.
 * However, there is a stronger problem with Trigonometric constants expressed in real radicals, which has multiple issues and contains many awful formulas that are WP:OR (that is, they cannot be sourced nor verified without a difficult computation). So, a better solution could be to expand the article as suggested above, and merging here the encyclopedic content of Trigonometric constants expressed in real radicals (that is, essentially, a table of the simplest radical expressions. If this solution is retained, a section Root_of_unity should be created with a main template redirecting here, and a short summary of this article. D.Lazard (talk) 09:00, 20 November 2021 (UTC)
 * I think that makes sense. I agree that the vast majority of Trigonometric constants expressed in real radicals should be deleted. I recently deleted a bit of it, and consolidated some of it into a common angles section. Do you mean that after the merge that page would just redirect here? I started a section on the talk page there about what content should be kept. Danstronger (talk) 15:14, 20 November 2021 (UTC)
 * I agree with the suggestiona done there. D.Lazard (talk) 15:41, 20 November 2021 (UTC)

Is there any backup of the original content of "Trigonometric constants expressed in real radicals" available somewhere ? Or is that data lost now ? — Preceding unsigned comment added by 98.203.170.58 (talk) 16:35, 23 January 2022 (UTC)
 * Follow the link, and look at the history of the page. D.Lazard (talk) 17:01, 23 January 2022 (UTC)

Missing exact values
There are many other nice trigonometric expression! By Galois theory, there is an expression using roots whenever the galois group of cos(alpha) is solvable, whereas the constructible angles are only those whose Galois group is a 2-group. For example, the article talks about cos(360/7) not being constructible, but cos(360/7) has galois group Z/3, and thus can be expressed by a third root, specifically

$$\cos \left( 2\pi/7 \right) =1/6\,{\frac { \left( 28+84\,i\sqrt {3} \right) ^{2/3}-2\,\sqrt [3]{28+84\,i\sqrt {3}}+28}{\sqrt [3]{28+84\,i \sqrt {3}}}} $$

Even nicer is:

$$\cos \left( 2\pi/9 \right) =\frac{\sqrt [3]{-4+4\,i\sqrt {3}}}{2}+2\,{ \frac {1}{\sqrt [3]{-4+4\,i\sqrt {3}}}} $$

Coming to think of it, $$\mathrm{Gal}\left(\mathbb{Q}\left(\cos\left(\frac{2\pi}{n}\right)\right)/\mathbb{Q}\right)$$ is always abelian (being a quotient group of a cyclotomic Galois group), hence solvable, so $$\cos\left(\frac{2\pi}{n}\right)$$ should always have an expression via radicals... It could be nice to mention this...

— Preceding unsigned comment added by 132.64.72.195 (talk • contribs) 7 June 2022 (UTC)


 * Why don’t you write something up and add it. You can describe Gauss’s algorithm for finding these, and maybe link to https://dl.acm.org/doi/pdf/10.1145/240065.240070. –jacobolus (t) 18:55, 8 June 2022 (UTC)

Propose to consolidate tables
Currently, we have two tables: Exact trigonometric values and Exact trigonometric values, the former being a subset of the latter. I propose to remove the former and move the latter up, making some changes in the text that precedes the table. - DVdm (talk) 10:20, 2 February 2024 (UTC)


 * There has been a lot of disagreement over the table. Originally as copied from another page it was "angles from 0 to 90 degrees that are multiples of 15, 18, or 22.5 degrees". Then @Danstronger shortened it to angles from 0 to 45 degrees. Then it got replaced by another table, again from 0 to 90 . Then DanStronger shortened it back to 0 to 45. Then Mgkrupa extended it to 0 to 360. Then DanStronger shortened to 0 to 90. Then Mgkrupa extended it again to 0 to 330, and moved it to a template Template:Trigonometric_Functions_Exact_Values_Table. Then DanStronger changed it back to 0 to 90. Then Mgkrupa added backed the extended table but at the end. Then @Headbomb de-templated the extended table. Then @Jacobolus did their recent edits such as the removal of 7.5 degrees (added to the template by @Mgkrupa) I should also mention my addition of 3 degrees which got undone by an IP.
 * The way I see it, there are a lot of questions we never developed a consensus on:
 * What should the range of the table be? The sensible options all start at 0, and then can end at 45, 90, or 360. The shorter ranges are more concise and avoid duplication, but the 360 is most useful if you have to quickly look up a value.
 * What should the precision of the table be? Generally any angle 2 pi/n can be written with square roots (i.e. is constructible) if n is a product of Fermat prime powers (2,3,5,17), and then as IP writes in the section above,, you can use radicals more generally to express any integer fraction. As far as sources, the most precise I found gives formulas for all angles to 3 degrees precision. But this has been opposed and Jacobolus apparently thinks that 7.5 degrees is not needed. (IIRC 7.5 is in a source too)
 * What should the columns be? - Actually this is fairly settled, nobody has opposed the six-function format or the listing of both degrees and radians.
 * What is the point of the table? Danstronger/Jacobolus seems to think it is something like "concisely and non-redundantly list simple expressions for constructible angles", for Mkgrupa/me it is more like "list exact trigonometric formulas so that the reader doesn't have to read the article and derive these formulas themselves."
 * Mathnerd314159 (talk) 23:04, 2 February 2024 (UTC)
 * The 7.5 degree entry is like 50% wider than any of the others, and breaks the entire table layout. It's not worth the space. Feel free to add those entries elsewhere in the article. –jacobolus (t) 00:29, 3 February 2024 (UTC)
 * As far as "breaking the entire table layout", that is really an issue with the new skin, that the table is too wide for its fixed width design. The old skin looks fine. Also there is a "full width" button in the corner which fixes the overlap. There is ongoing debate about full width vs. fixed width and I suspect that the Web team is thinking about ways to further improve the situation.
 * As far as "50% wider than any of the others", that is because there is only 7.5 currently. If we added the entries for 3,6,9,etc., then the table would be consistently big. I have thought about doing so but I am worried about putting in a lot of effort to transcribe the formulas from the sources only to see it reverted.
 * As far as "not worth the space", WP:NOTPAPER.
 * And adding them elsewhere in the article, the problem is that there are two tables with substantially the same content. You are saying, make a third table with the extra-wide formulas? Absurd. Mathnerd314159 (talk) 02:59, 3 February 2024 (UTC)
 * No, it look terrible in any skin. The formulas are simply too wide, like 50% wider than any other row, and there's no obvious way to narrow them. Cramming them into this table is hostile to readers. Adding even more rows that are even wider still is not going to solve the problem.
 * If you want to include these or other very complicated formulas, you should put them as ordinary blocks of math in the appropriate section of the article, instead of trying to force them into a table where they simply don't fit. Alternately you could try making a new table that just shows sine and cosine, or maybe sine, tangent, and secant; if you cut down to 2 columns you can fit something wider than when there are 6 columns.
 * If you have 6 columns of formulas you are limited to about 6 or 7em width per column before the layout starts breaking on a substantial proportion of reader's displays. Maybe you could push one or two columns slightly wider at the expense of making the table moderately less legible, but the specific entries in question have a pair of ~16em wide cells, which are just absurdly too big to fit.
 * If your concern is what Wikipedia is not, check out WP:NOTDATABASE. There are infinitely many rows we could conceivably include (see for an infinite list of the polygons whose vertices we could describe in this table), and we could also show arbitrarily many columns, if you want including haversine, chord, half-tangent, etc. If we have to pick some criteria for limiting the selection, "what can render in a way that readers can actually read it" is much better than most. –jacobolus (t) 03:03, 3 February 2024 (UTC)
 * Well these look OK:, . Maybe it is because they use 1-2 columns and really only do sine. But I don't see why a table of sin/cos/tan/cot/sec/csc would be any less readable. I think when you have values 0-90 and the values of sin and so on then writing them in a table is pretty natural - I don't see how writing "ordinary blocks of math" is going to work. Do you write paragraphs like:
 * 1:
 * 2:
 * ? Mathnerd314159 (talk) 03:21, 3 February 2024 (UTC)
 * Take a look at the "Derivations of constructible values" section for what I mean by block math outside a table. We could add another section (not sure the right title) listing just the sines, or a few trig functions, for 2π/n for n the first several values of, as ordinary paragraphs, or if there's no additional relevant text to add, in a table with only ~2 columns. Your first link is incredibly ugly and hard to read, let's not do that one. Your second link is fine; notice this is not a table. –jacobolus (t) 03:28, 3 February 2024 (UTC)
 * Thanks for starting a discussion about the tables, it's interesting that they're so controversial. (On the topic of 7.5 degrees, at least the cosine is there (as pi/24) in the Half Angle Formula section.)  To summarize my objections to the extended table (as it is, or extending it further):
 * It becomes an indiscriminate collection of information. I suspect it would be exceedingly rare that someone would ever actually need the expression for sin(9 degrees) or cot(285 degrees).  Why not 3/8 of a degree or 1083 degrees.
 * More specifically, the values over 90 are redundant. I could see having multiples of 30 and 45 degrees up to 360 to illustrate the signs of the different functions, but anything more than that seems unnecessary. On the off chance that someone really does need to know cot(285), I'm pretty sure they would be able to figure out that it's -cot(75).
 * A lot of the values currently there, like sin(15), are written unsimplified. Is there something instructive about factoring a sqrt(2) out of sqrt(6) + sqrt(2)?  I don't think so, but if there is maybe we should explain what it is.
 * Danstronger (talk) 06:31, 3 February 2024 (UTC)
 * There's no inherently clear definition of "simplified". They don't have to be in any particular form, but they should all be consistent in a table; previously it was a mishmash. –jacobolus (t) 16:19, 3 February 2024 (UTC)
 * Article titled "Exact trigonometric values" should list "Exact trigonometric values". Mgkrupa  06:48, 3 February 2024 (UTC)
 * A LOT of people who are just beginning to study trigonometry and/or algebra use this article. So let me remind everyone about some of the Wikipedia guidelines for math articles.
 * What assumptions can and can not be made about this article and its "typical" reader. According to Manual of Style/Mathematics, and Make technical articles understandable, and Manual of Style this article should follow the following guidelines (as well as others not listed here). It should be be written "one level down", which means:
 * "'Write one level down": "consider the typical level where the topic is studied (for example, secondary, undergraduate, or postgraduate) and write the article for readers who are at the previous level.
 * Writing one level down also supports our goal to provide a tertiary source on the topic, which readers can use before they begin to read other sources about it."
 * "articles on undergraduate topics can be aimed at a reader with a secondary school background, and articles on postgraduate topics can be aimed at readers with some undergraduate background."
 * "Articles should be as accessible as possible to readers not already familiar with the subject matter."
 * It is safe to assume that a reader is familiar with the subjects of arithmetic, algebra, geometry"
 * For articles that are on these subjects, or on simpler subjects, it can be assumed that the reader is not familiar with the aforementioned subjects."
 * "Any topics outside of that scope or more advanced than them a reader can be assumed to be ignorant of."
 * The point is that you should not assume that what is easy for you is easy for the reader.
 * In particular, you should (and are required to) take into account that many, many readers of this article do NOT consider it trivial to deduce trig function values for angles between 90° and 360° from the trig values for $$0^\circ \leq \theta \leq 90^\circ.$$ Keep in mind that many readers find this difficult. Consequently, if the table is limited to angles $$0^\circ \leq \theta \leq 90^\circ,$$ they might leave without the information they came for or worse: they may leave with the wrong value because they made a mistake while attempting to deduce the value.
 * This is why the article should keep the extended table for all $$0^\circ \leq \theta \leq 360^\circ.$$
 * Those editors who think the extended table - which is at the very end/bottom of the article - is too large are free to ignore those rows for the benefit of those many readers who still find this subject matter difficult.
 * Another option is to split the extended table in 2: the first table for angles $$0^\circ \leq \theta \leq 90^\circ$$ (I wouldn't object to removing this first table since already appears at the top of the article) and the second for angles between 90° and 360° and then include in-between these tables an explanation of how the values in the second table can be obtained from those in the first.  Mgkrupa  22:50, 3 February 2024 (UTC)
 * A LOT of people who are just beginning to study trigonometry and/or algebra use this article. – I certainly hope not. The material in this page is reference trivia almost entirely irrelevant to people just beginning to study trigonometry. They should instead be looking at trigonometry, trigonometric functions, unit circle, etc. (all of which should ideally be dramatically improved). –jacobolus (t) 23:02, 3 February 2024 (UTC)
 * The Scott Surgent source has MAT170 in the URL which I believe is Precalculus. The James T. Parent source is a math professor at a community college, and I would assume he too prepared that table for his calculus or precalculus students. So yeah, the typical reader at this level is learning trigonometry and is most likely not very familiar with all of the laws (shifts, reflections, etc.). I can also support this from my personal recollection: when I was learning about the sine of 15 degrees it was around the time I was finishing up algebra. Admittedly it was at a math summer camp (MathPath), but I can imagine that there are many curious readers who will click through from those pages you linked like Trigonometric functions if they want to see a worked out list of expressions for all the angles.
 * Now as to whether the values of all the angles are relevant, well, it is true that for practical purposes 30 60 90 are all that anyone remembers, and people can use calculators for the rest. But these angle formulas could appear in math competition problems and the like. The relevant standard I think is WP:NLIST (a table is a form of list, right?). I have linked several sources which give a fair amount of formulas so I think it is notable. So if it doesn't fit here I could create a new page Table of exact trigonometric values with the full table. Mathnerd314159 (talk) 02:08, 4 February 2024 (UTC)
 * Surgent's page is in the "Other mathy things" part of his personal website. It's certainly nothing required by any "precalculus" curriculum (how many precalculus students know what a minimal polynomial is?), and there's no indication he expects introductory students to be using it as a reference for their homework. You might think of it (and this wiki article for that matter) as largely a tangential supplement for people who are curious to learn something beyond the school curriculum. (And you'll notice it only covers the range 0–45°.) A starting student doesn't really need any information not included in a picture along the lines of:
 * Unit_circle_angles_color.svg
 * And frankly for nearly all student purposes this could be cut down to the first quadrant.
 * If the goal were only to make something as useful as possible for students taking a high-school-level trigonometry course, we should try to make the tables contain as few rows and columns as possible, because every bit of extra information imposes some cognitive burden. They certainly aren't going to be looking up a closed formula for the exsecant and haversine of 296°&hairsp;15′ or whatever. –jacobolus (t) 03:50, 4 February 2024 (UTC)
 * Frankly, I know almost nothing about Wikipedia editing, but as a current trigonometry student, I have found the extended table of values extremely useful. Could I figure out the correct values of trig functions from 90°-360° using just 0°-90°? Yes, of course, but it has been much easier for me to just look it up in this table. You are right that no one would ever look up the value of haversine 296°15′, so, obviously, we shouldn't include that. However, values like 135° are much more likely to be needed and should be included. Catopine (talk) 02:44, 28 February 2024 (UTC)
 * Have we found many (or any) independent reliable sources for the values for these less common angles (or angles beyond 90 degrees)? Blogs aren't reliable sources, even if the author is a math professor, and we shouldn't be checking their work or figuring it out for ourselves.  If good sources can be found, they may help us figure out where to draw the line of which values to include. Danstronger (talk) 01:48, 5 February 2024 (UTC)
 * It's a good point, here are some old books:
 * Cagnoli, 1808, French, sine 3-90 by 3
 * French, 1888:, gives sine, cosine, secant, cosecant, tangant and cotangent of multiples of 3 degrees up to 90. On page 58 there is a note "This is the first time, we believe, that the values with rationalized denominators of the secants, cosecants, tangents, and cotangents of all the multiple arcs of 3° have been given"
 * French sine 0-90 by 3
 * 1833, sine 0-90 by 3
 * 1855, sine 0-90 by 3
 * . sine/cosine 0-90 by 3. Cites "Cagnoli's Trigonométrie". There is a secondary table using a notation where 100° is a right angle, with multiples of 10 up to 90 - had me confused for a bit.
 * sine 0-90 by 3
 * sine 0-90 by 3
 * sine of 7.5 and some other values
 * German sine/cosine of 0-90 by 3
 * German sine/cosine of 0-90 by 3
 * German sine/cosine of 0-90 by 3
 * German sine/cosine of 0-90 by 3
 * German, sine of values 3-87 in a weird order
 * German, sine of values 3-87 in a similar weird order
 * German sine/cosine of 0-90 by 3
 * German sine of 0-90 by 3, in a weird order
 * Japanese sine 3-87 by 3
 * Latin, sine 0-90 by 3
 * There are probably more but I was in page 20 of a Google Books search for  and there were a lot of irrelevant results and duplicates so I decided to stop. Worth noting that all of the results seem to be 1800's - that could just be what Google Books indexes well, or maybe that is the era that these formulas were popular.
 * I would say that given that the common denominator is sine 0-90 by 3, that should definitely be included, and since there are sources for 7.5 degrees and the cosine, secant, etc. that those should be included too. But it seems the extended table 90-360 is probably not a good idea. Mathnerd314159 (talk) 04:36, 5 February 2024 (UTC)
 * Interesting. I think that's pretty good support for us having sines of angles that are multiples of 3 degrees or 7.5 degrees (instead of the extended table). To avoid redundancy my suggestion would be just sines from 0 to 90, or (probably better) sines and cosines from 0 to 45. (To avoid redundancy even more aggressively, we could label this section with "other angles divisible by..." and exclude the ones already present in the first table.  So there is just 3, 6, 7.5, 9, 12, 21, 24, 27, 33, 37.5, 39, and 42.) And probably a list is better than a table for these.  I like the layout at [], with the shorter sines and cosines at two per line and the longer ones at one per line.
 * This could go in between the derivations and the 17 section, with the segue being that these multiples of 3 and 7.5 degrees are all derivable with the above methods. Danstronger (talk) 06:24, 5 February 2024 (UTC)
 * It could alternately go in sections like: 15-gon, 16-gon, 17-gon, 20-gon, 24-gon, 30-gon, 32-gon, 48-gon, 60-gon, 120-gon (3°), only showing the novel values at each section, maybe sines and cosines up to 45° or just sines up to 90°. –jacobolus (t) 06:33, 5 February 2024 (UTC)
 * It could alternately go in sections like: 15-gon, 16-gon, 17-gon, 20-gon, 24-gon, 30-gon, 32-gon, 48-gon, 60-gon, 120-gon (3°), only showing the novel values at each section, maybe sines and cosines up to 45° or just sines up to 90°. –jacobolus (t) 06:33, 5 February 2024 (UTC)

I agree that the values over 90 are redundant. So here's my second proposal. Let's just keep the current table (up to 90 degrees) at the top, and dump the extended table. - DVdm (talk) 11:43, 3 February 2024 (UTC)
 * Seems like a fine idea. Instead of the "extended" table, it might be better to put just the sine/cosine of a wider set of "small" angles in a new section, e.g. π/12, π/15, π/16, π/17, π/20, π/24, π/30, π/60. We could move the 17-gon subsection down there, as the current text about it doesn't really match the higher-level heading about "derivations". –jacobolus (t) 16:16, 3 February 2024 (UTC)
 * That sounds fine to me. Danstronger (talk) 00:45, 4 February 2024 (UTC)
 * Actually, I have another idea. What if we change the title of "Derivations of constructible values" to a more generic "Specific values", and then extend it with more subsections, starting from the multiples of π/2 and working our way down. At a basic level we can include information about the sine, cosine, decimal approximation, minimal polynomial, etc., but can also include some derivation/construction, a diagram ("proof without words"), historical discussion, or whatever else is relevant to each particular entry. This section could conceivably grow substantially so long as there are enough sources to support discussion of each particular value added. –jacobolus (t) 21:26, 4 February 2024 (UTC)
 * Yeah, I was trying to think of a good way to circumscribe an additional list of values or content, so that it doesn't become bloated. I think it makes sense to restrict the content to stuff that is well-sourced (not just someone's blog) and where it's illustrating a point or at least where the reason that it's interesting is clearly explained. Danstronger (talk) 00:50, 5 February 2024 (UTC)
 * Why not both? Mgkrupa  19:34, 4 February 2024 (UTC)
 * I implemented the edit (splitting the table) that I talked about above. Your opinion? Mgkrupa  19:48, 4 February 2024 (UTC)
 * Two tables is definitely one too many. - DVdm (talk) 20:07, 4 February 2024 (UTC)
 * Now (the visible-by-default part of) the extended table is exactly the same as the original table, except that the values are simpilfied more verbosely. Danstronger (talk) 00:52, 5 February 2024 (UTC)
 * The visible-by-default table would be removed. Mgkrupa  23:27, 14 February 2024 (UTC)

"Mathematics and Statistics" Horizon Research Publishing source
You can see the discussion here: https://web.archive.org/web/20160421041423/https://scholarlyoa.com/2013/05/02/horizon-research-publishing-corporation/. Beall really doesn't have any dirt, it is just he got the "spam" advertising the new journals and he added it to the list because it was OA. Later on he admits "It is possible that some respected authors have published in the journal." Mathnerd314159 (talk) 06:59, 24 February 2024 (UTC)


 * @Headbomb Where is this evidence you mention? Mathnerd314159 (talk) 21:07, 26 February 2024 (UTC)
 * Here is a paper arguing that it is not a predatory publisher: https://www.iaras.org/iaras/filedownloads/ijc/2022/006-0009(2022).pdf Mathnerd314159 (talk) 21:23, 26 February 2024 (UTC)