Talk:Trigonometric constants expressed in real radicals

message: In the article, it says that cot 0˚ has "no solution". Should this be instead written "undefined"? Is there a difference? User:Error792
: I don't think there's a difference. Tom Ruen 16:07, 23 June 2007 (UTC)
replies: text: "Undefined" means "no solution in general." "No solution" precludes a mathematical context in which cos 0 is meaningful. That's why x/0 is undefined - I'm sure you can already think of a meaningful interpretation of it, but that depends on the context in which you construct x/0. If you think of it a different way - in a different mathematical context - you can arrive at a contradictory interpretation. ᛭ LokiClock (talk) 13:06, 27 May 2010 (UTC)

text: For a "solution" you need a "problem". There is no problem; cotangent is a function, and zero is not in its domain, so cot(0) doesn't mean anything. Saying that cot(0) is undefined is more appropriate. Mike Pierce 3 October 2021  — Preceding undated comment added 15:12, 3 October 2021 (UTC)
thread_title:  "No Solution" 

message: :
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
The last two expressions can be shown to be the same.  Square the second one, and factor.
And my calculator, evaluates all three to the same value.
:--MathMan64 02:02, 2 August 2005 (UTC)
replies: text: I fixed this section to show how and when nested radicals can be simplified.--MathMan64 02:28, 8 September 2005 (UTC)
thread_title:  Is it the same value? 

message: Yes I left out a factor of (y - 1).  Thanks for finding it.
::--MathMan64 20:17, 8 March 2006 (UTC)
thread_title:  Factoring a fifth degree 

message: I have heard that a formula for the sine of one third of an angle can be derived from the half angle and angle addtion indentities. It is something along the lines of 4sin(3x)^3-3sin(3x)-sin(x)=0. Does anyone know what the real formula is and should it be put in?Hiiiiiiiiiiiiiiiiiiiii 00:04, 2 May 2006 (UTC)
replies: text: My assumption has been that this can't be solved in a radial form, but I've not looked into this. Tom Ruen 00:41, 2 May 2006 (UTC)
replies: text:  Well if there is a cubic equation with nontranscendental coefficients, it can be solved with radicals.Hiiiiiiiiiiiiiiiiiiiii 00:10, 4 May 2006 (UTC)
replies: text:  Sounds like a good challenge - to be proven or disproven. I started this article from math notes I kept from 20 years ago in my Trig class extra credit, never saw them printed up in any books. I'm happy if it can be shown to go further using closed cubic polynomial solutions. The nice thing about these are that they are pretty easy to test numerically on a calculator to confirm correctness. Tom Ruen 02:00, 4 May 2006 (UTC)
replies: text:  Well I found this on another wikipedia page. sin(3x)=3sin(x)-4(sin(x))^3. It appeared to be correct when I tested it with a calculator. Hiiiiiiiiiiiiiiiiiiiii 15:24, 6 May 2006 (UTC)

text:  Okay, I see it at Trigonometric_identities#Triple-angle_formulae. Next step, solve that cubic equation for sin(x)=f(sin(3x)). Then test on something simple like sin(15)=f(sin(45)). If it works, you can work on sin(11), sin(10), sin(8), sin(7), sin(5), sin(4), sin(2), sin(1), etc, in any order you like. Tom Ruen 08:45, 22 May 2006 (UTC)
replies: text:  Actually, I'm working on this problem right now. The first two roots of the equation are the imaginary ones, so I'm going to start on the third. The speed of my work depends on my laziness, so I expect to have this done within the next week. CodeLabMaster 17:00, 6 June 2006 (UTC)
replies: text:  See my comment in the pi/7 section.  76.199.137.78 (talk) 03:43, 6 December 2007 (UTC)
thread_title: Sin(1 degree)

message: I was doing some work with regular heptagons,  which I now read are not ruler-and-compass constructible,  and got to the fact that the cosine of pi/7 satisfies the cubic equation -
x^3 - x^2 - 2x + 1 = 0
where x = 2cos(pi/7),  which is the ratio of the shorter diagonal to the sidelength in a regular heptagon.
Curiously enough,  the ratio of the sidelength to the longer diagonal,  which is sin(pi/7) / sin(3pi/7),  is also a solution of this cubic - but not the same one.
I don't know if this means that the trig ratios for multiples of pi/7 should be included in this article or not.  In a sense,  this is an exact mathematical specification for the ratio as opposed to an approximation.  On the other hand,  from what I've read (today on wikipaedia) about solving cubics,  I gather that actually computing these values requires reference back to trigonometric ratios anyway. - there is no expression in terms of surd-like expressions.
A useful reference is:
http://mathworld.wolfram.com/TrigonometryAnglesPi7.html
which gives sin & cos (and others) for pi/7, 2pi/7, 3pi/7 in terms of roots of polynomials.
Ben D R 08:19, 23 January 2007 (UTC)
replies: text: Actually, a cubic (and even a quartic) can be solved in radicals, though it's hairy enough that usually nobody bothers to do so.  So it would be possible to include explicit formulas for trig functions of arbitrary multiples of both pi/360 and pi/7.  But they're still not constructible with straightedge and compass, because that only gives you a way to do square roots, not cube roots.  The constructible numbers are a proper subset of the expressed-in-radical numbers, which are a proper subset of the algebraic numbers.  76.199.137.78 (talk) 03:43, 6 December 2007 (UTC)
thread_title:  pi / 7 

message: How about showing how these constants can be calculated using the series definitions of the functions? That I would like to see. 163.1.62.82 (talk) 20:40, 13 January 2008 (UTC)
thread_title:  Series calculations 

message: So do we want to start on some of the non-constructible angles such as 20 degrees?
:\sin \frac {\pi}{9} = \sin 20^\circ = 2^{-\frac{4}{3}} \left( \sqrt[3]{i-\sqrt3} - \sqrt[3]{i+\sqrt3} \right) 
See http://mathworld.wolfram.com/TrigonometryAnglesPi9.html
::--MathMan64 (talk) 19:36, 20 February 2008 (UTC)
thread_title:  Sine 20 degrees 

message: :Wolfram´s Formula is correct, but I wonder why Euler formula is not applied as given below:
\cos 20^\circ =  \cos \frac {\pi}{9} =\frac {e^{+i\frac{\pi}{9}}+ e^{-i\frac{\pi}{9}}}{2} = \frac { \sqrt[3]{e^{+i\frac{\pi}{3}}}+ \sqrt[3]{e^{-i\frac{\pi}{3}}}}{2} 
\cos 20^\circ =  \frac { \sqrt[3]{\cos 60^\circ + i\sin 60^\circ }+ \sqrt[3]{\cos 60^\circ -i\sin 60^\circ }}{2} 
\cos 20^\circ =   \frac { \sqrt[3]{1+i\sqrt 3}+ \sqrt[3]{1-i\sqrt 3}}{2\sqrt[3]{2}}   
\sin 20^\circ =  \sin \frac {\pi}{9} =\frac {e^{+i\frac{\pi}{9}}- e^{-i\frac{\pi}{9}}}{2i}= \frac { \sqrt[3]{i- \sqrt 3}- \sqrt[3]{i+\sqrt 3}}{2\sqrt[3]{2}}  
----
In order to facilitate remembering as well as non-scientific pocket calculator routine  exact constants for k×3°/16 ≤ 45° (k  ≤ 240 is whole namber) should be expressed in common pattern: 
\sin\frac{k*3^\circ}{16}=\frac{1}{2}\sqrt{2-f_k} \mbox{ and }  \cos\frac{k*3^\circ}{16}=\frac{1}{2}\sqrt{2+f_k} \mbox{ consequently}  
 \tan\frac{k*3^\circ}{16}=\sqrt{\frac{2-f_k}{2+f_k}}=  \cot^{-1}\frac{k*3^\circ}{16}  \mbox{ where } f_k=2\cos\frac{k*3^\circ}{8}=f(\varphi,k)  
sin(2×36°) = sin72° = sin108° = sin(2×36° + 36°) or  2sin36°cos36° = (2sin36°cos36°)cos36° + (2cos236° – 1)sin36°  that divided with sin36° gives  
(2cos36°)2 –  2cos36° – 1 = 0 i.e. quadratic for 2cos36° = φ (golden ratio or Fibonacci number) along with its positive root: 
 2\cos36^\circ = \frac{1+\sqrt{1^2+4}}{2}=\frac{\sqrt{5}+1}{2}= \varphi = 1+\frac{1}{\varphi} = \varphi^2 -1 \Rightarrow   
  
  \begin{align}
  & \sin36^\circ \\ 
 & \cos36^\circ \\ 
\end{align} = \frac{1}{2}\sqrt{2 \mp  (\varphi -1) } \mbox{ and } \begin{align}
  & \sin18^\circ \\ 
 & \cos18^\circ \\ 
\end{align} = \frac{1}{2}\sqrt{2 \mp  2\cos36^\circ } = \frac{1}{2}\sqrt{2 \mp  \varphi }  
 
Supposing that the constants for 0°, 45°, 30° and 15° = 30°/2 are well known, it is sufficient to apply additional formulae only for cos(18° ± 15°) = cos33°; cos3°:     \begin{align}
  & \cos3^\circ \\ 
 & \cos33^\circ \\ 
\end{align} = \sqrt{(\cos18^\circ \cos15^\circ \pm \sin18^\circ \sin15^\circ )^2} = \frac{1}{2}\sqrt{2+\frac{\varphi\sqrt3 \pm  \sqrt{3-\varphi}   }{2}}  
Double-angle and half-angle formulae should be preferred for other angles: 6° = 2×3°, 12° = 2×6°, 24° = 2×12°, 42° = 90° – 2×24°, 9° = 18°/2, 39° = (90° – 12°)/2 , 21° = 42°/2, 27° = (90° – 36°)/2: 
After squaring and square rooting of fk exact constants for all multtiples of 3° can be united into single algebraic function:
 \begin{align}
  & \sin(n*3^\circ) \\ 
 & \cos(n*3^\circ) \\ 
\end{align} = \frac{1}{2}\sqrt{2 \mp \sqrt{2+ \frac{t}{2\varphi^u}+\frac{v\varphi^{uw}}{2\varphi^u}\sqrt{9+3w(u\varphi^u-1)}} }    \mbox{ where }  \varphi = \frac{\sqrt{5}+1}{2} \approx 1.6180339887498948482045868343656 ... and unary operators t, u, v and w are  depending upon 0 ≤ n ≤ 15 as follows:   
  t(n)=\frac{\cos(n*36^\circ)}{|\cos(n*36^\circ)|} = \pm 1, u(n)= \sin^0(n*72^\circ)\frac{\cos(n*72^\circ)}{|\cos(n*72^\circ)|} = 0; \pm 1,     v(n) = \frac{\cos(n*12^\circ)}{|\cos(n*12^\circ)|} = \pm 1,w(n) = \sin^0(n*72^\circ)\sin^0(n*120^\circ) = 0;1,    Obviously the table at main article could (should?) be substituted by last five lines! 
Repeating half-angle formula we get fk fork×1°30´ = k×3°/2, k×45´ = k×1°30´/2, k×22´30˝ = k×45´/2 and k×675˝ = k×11´15˝ = k×22´30˝/2:
f_k = 2\cos\frac{k*3^\circ}{2^3} = \sqrt{2+\sqrt{2+\sqrt{2 + \sqrt{2+2\cos(k*3^\circ)}}}}; \, k=1 \Rightarrow  
\delta = \sin\frac{3^\circ}{2^4} = \frac{1}{2}\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\frac{\varphi\sqrt3 + \sqrt{3-\varphi}}{2}}}}}}  
δ = sin675˝ ≈ 0.003272486506526625459989844900576... marginal value of Interpolation formula
 
77.238.200.190 (talk) 06:38, 4 October 2008 (UTC)
thread_title:  Sine 20 degrees and exact values for every 675 seconds 

message: I have seen analytical expressions of sine of 1, 10, 20 degrees, etc. that are in complex format whose real and imaginary parts cannot be analytically separated. If we accept this kind of format, we might have much simpler solution for all integer degrees from 1 to 89 as follows:
\sin(n^\circ)=\mbox{Im}(i^\frac{n}{90}),
\cos(n^\circ)=\mbox{Re}(i^\frac{n}{90}).
where n is any integer from 1 to 89.
--Roland 06:46, 8 May 2015 (UTC)
replies: text: Sorry, I don't think this will help, looks like a circular definition. Tom Ruen (talk) 18:02, 8 May 2015 (UTC)
replies: text:  My point is that other complex expressions that contain i have no more meaning than the above much simpler ones. --Roland 01:53, 9 May 2015 (UTC)
thread_title: sine 1 degree, sine 10 degree, sine 20 degree, etc. 

message: In the section on simplifying nested radicals, the final expression appears to be equivalent to 
:  \pm\sqrt{\frac{a\pm R}{2}} \pm\sqrt{\frac{c(a\pm R)}{2c}} 
and I'm not convinced that this is correct ... —Tamfang (talk) 19:33, 7 April 2008 (UTC)
thread_title:  nested radicals 

message: The section "Fermat number" says 
:the trigonometric functions of other angles, such as 2π/7, 2π/9 (= 40°), and 2π/13 ... are soluble by radicals.
and the context implies that these formulas would be practical. Actually the one for 2π/7 is not practical, because it involves solving an irreducible cubic, which is solvable in radicals of complex numbers but not solvable in radicals of real numbers. As for 2π/9 and 2π/13, finding their sines or cosines might require solving respectively an 8th degree and a 12th degree equation; such equations cannot in general be solved in radicals at all unless the polynomial is reducible with a quartic or lower factor, which I doubt. Or maybe they only require solving 4th and 6th degree equations respectively, the former of which would probably lead to cube roots of non-reals and the latter of which would probably not be solvable in radicals.
So I suspect that the above statement is false. Does anyone have a cite that says they are in fact solvable in radicals (preferably real radicals)? Or, does anyone mind if I delete the above passage?  Duoduoduo (talk) 22:20, 9 March 2012 (UTC)
replies: text: The trigonometric functions of π/7, π/9, π/11, and π/13 can all be solved using radicals. The first two only need cubics. The next one needs a quintic and the last a sextic, but luckily the equations involved happen to be among those that can be solved in radicals. The relevant info can be found on MathWorld. Double sharp (talk) 06:03, 19 February 2015 (UTC)
thread_title: Dubious assertion

message: The diagram showing Ptolemy's theorem in the section "Calculated trigonometric values for sine and cosine" has line length label errors.  I will try to remedy that unless someone else gets in before I do.Siw1939 (talk) 14:24, 27 July 2014 (UTC)
thread_title:  Ptolemy theorem diagram 

message: Can you compute sin\left(\frac{2\pi}{2^n+1}\right), cos\left(\frac{2\pi}{2^n+1}\right) for every positive integer n?
These values should have \sqrt{2^n+1}, for example, when n = 5, the values are \frac{\sqrt{10+2\sqrt{5}}}{4} and \frac{\sqrt{5}-1}{4}.  [typo--the poster meant n=2 so 2^n+1=5.\,]
:No--e.g., \cos \frac{2\pi}{2^3+1} = cos 40° cannot be expressed in square roots. Loraof (talk) 16:59, 24 September 2015 (UTC)
thread_title:  values for 2pi/(2^n+1) 

message: The current title of the article is "Exact trigonometric constants". By this is meant "trigonometric constants expressed in real radicals", the implication being that e.g., \cos 60^\circ is somehow not an exact expression. That's wrong—in fact expressions involving trigonometric functions are exact in the same way as are expressions in radicals. So I'm moving the page to the new title "Trigonometric constants expressed in real radicals. Loraof (talk) 18:22, 24 September 2015 (UTC)
thread_title:  Reason for article move 

message: In the list of trigonometric constants to 2 pi / n where n=7, the given formula includes the value omega with no apparent definition of omega. Can anyone provide the definition of omega?  — Preceding unsigned comment added by 2601:C2:8300:3011:D515:F39D:1321:D7DB (talk) 21:14, 5 October 2017 (UTC)
replies: text: Done. Thanks for pointing this out. Loraof (talk) 17:58, 15 October 2017 (UTC)
thread_title:  Omega is undefined in the list of trigonometric constants to 2 pi / n where n=7 

message: In this section several formulas involve cubic roots of complex numbers. Such a cubic root is well defined only if one consider the principal value of the cubic root, which is the cubic root with the largest real part. In fact, if one changes the choice of the cubic root, this changes the value of the whole formula for providing either a non-real result if the two cubic roots appearing in the formula are not changed accordingly, or provides the value of the sine or cosine of an angle larger than 2\pi/n.
The formulas for the cosine that contain all a sum of cubic roots of two conjugate numbers. They are thus correct (for the choice of the cubic roots), because changing the cubic roots from the principal value to another value would decease the cosine. On the other hand, for sine, one must choosing the cubic roots for getting the smallest final value. Therefore the values of sine for n=7,9,18 are wrong, and I'll change them. D.Lazard (talk) 11:11, 13 November 2018 (UTC)
thread_title:  Cubic roots in section "List of trigonometric constants of {{sfrac|2''&pi;''|''n''}}" 

message: I see the line:
\frac{i}{4}\left(\sqrt[3]{4-4\sqrt{-3}}-\sqrt[3]{4+4\sqrt{-3}}\right)
Should this be 
\frac{1}{4}\left(\sqrt[3]{4-4\sqrt{-3}}-\sqrt[3]{4+4\sqrt{-3}}\right)
ThaniosAkro (talk) 22:42, 11 February 2019 (UTC)
replies: text: The expression with the 'i' seems to be correct. I checked this by evaluating in my computer algebra system. There really should be a verifiable source for all of these expressions though...  MaxwellMolecule (talk) 23:03, 11 February 2019 (UTC)
replies: text: Here is an easy proof that 1 instead of  is wrong: the cube roots are complex conjugate. Thus, their difference is purely imaginary, and one has tp multiply by  for getting a real result. D.Lazard (talk) 10:35, 12 February 2019 (UTC)
thread_title:  i instead of 1? 

message: In the introduction I see:
For an angle of an integer number of degrees that is not a multiple 3° 
Would it be better to say:
For an angle of an integer number of degrees that is not a multiple of 3° 
ThaniosAkro (talk) 17:49, 19 February 2019 (UTC)
thread_title:  Small grammatical error? 

message: All angles should be given in radians. (Angles in degrees within parentheses). Won't spend time doing it, since it will likely be reverted.  — Preceding unsigned comment added by 142.58.132.230 (talk) 05:52, 22 February 2019 (UTC)
thread_title:  Angles should be given in radians 

message: It seems that most of the content here is original research or unencyclopedic.  At Talk:Trigonometric_number, User:D.Lazard suggests merging content from here to there.  I think this makes sense.  There is a lot to sift through here, but I would think the content that should be kept and moved to Trigonometric number would be 1) a discussion of the relationship between expressibility in radicals and constructibility and the condition for the angle to be constructible, 2) the table (and maybe derivation of sin(18)) at Common angles, and 3) an explanation of how you would derive some of the other values, with a few examples.  And then this page could just redirect there. Danstronger (talk) 15:04, 20 November 2021 (UTC)
replies: text: I agree with these suggestions. D.Lazard (talk) 15:39, 20 November 2021 (UTC)

text: "Trigonometric number" is a neologism only ever used by one obscure source. It seems like a weird thing for Wikipedia to canonize with an article. –jacobolus (t) 04:59, 25 November 2021 (UTC)
replies: text: Do you have a better title? The subject of this article is Real part of a root of unity; this seems a too long title. D.Lazard (talk) 08:23, 25 November 2021 (UTC)
replies: text:  I feel like the notable concept is the idea of expressing these numbers in terms of square roots and the resulting expressions.  "Trigonometric constants expressed in real radicals" sort of gets it right, but it's way too long. How about exact trigonometric values? This seems to me like a popular search phrase when I type various things into google. And it would remove the awkwardness of having to carve out the rational numbers.
Danstronger (talk) 12:04, 25 November 2021 (UTC)
replies: text: "Exact" is meanningless and confusing, as cos(1/\pi) it a trigonometric value that has an exact value, even if it is probably a transcendental number. A better title would be Real cyclotomic integer. In fact, for every rational number , the number 2\cos(2\pi r) is a real algebraic number that belongs to a cyclotomic field. Such a number is commonly called a real cyclotomic integer (Scholar Google provides more than 400 hits for "cyclotomic integer", and two articles that have "real cyclotomic integer" in their title). Moreover, every real cyclotomic integer has the form \sum_{k=1}^n 2 a_k\cos(2k\pi/n) with a_k integer. So, I suggest to keep the current title for the moment, and to move the article to Real cyclotomic integer when these notable properties will be added to the article (they must be added). D.Lazard (talk) 16:38, 25 November 2021 (UTC)

text:  These are also Chebyshev nodes (of the second kind). Aside: is there a reason the trigonometric number article claims 0, 1, 1/2 should be excluded? –jacobolus (t) 17:42, 25 November 2021 (UTC)
replies: text:  That's because Nivens defined "trigonometric number" so that it's required to be irrational. I think a page should be added for "cyclotomic integer" separate from this one (since most cyclotomic integers are not trigonometric numbers, but the information about how to derive sin(18) etc. would not be appropriate for a page on cyclotomic integers). And the connection between trigonometric numbers to both cyclotomic integers and chebyshev nodes should be noted on this page (like the connections to roots of unity and constructibility).  But it's probably better if most of the information on this page -- the square root expressions and the methods for deriving them -- are on a relatively accessible page, i.e. one with a title and lead that high school students can understand.  I still like exact trigonometric values best (exact means it's an algebraic expression instead of the first few decimals), but I don't hate trigonometric number (even though I agree that that concept is not really notable). How about algebraic expressions for trigonometric numbers?  I could even see something like Trigonometric constants expressed with square roots (I'm not a fan of "real radicals"). Danstronger (talk) 18:17, 25 November 2021 (UTC)
replies: text: Ok for keeping separate articles. I suggest Algebraic trigonometric value, as "algebraic" is clearer than "exact". D.Lazard (talk) 18:40, 25 November 2021 (UTC)
replies: text: Hmmm, but then "algebraic" sounds like it means algebraic. Danstronger (talk) 03:16, 26 November 2021 (UTC)
replies: text: Indeed, "algebraic" is correct, as trigonometric constants are algebraic numbers. D.Lazard (talk) 09:38, 26 November 2021 (UTC)

text: According to All Math Words Encyclopedia (McAdams, David E. 2nd Classroom edition, p 74.) these are called "exact values of trigonometric functions", sometimes also  "analytic values of trigonometric functions" and "trigonometric values of special angles". This no-merge merge was done too soon, by the way. A four-day "discussion" does in no way reflect consensus on the future of Trigonometric constants expressed in real radicals (1800 monthly views) and Trigonometric number (400 monthly views). The merger, Danstronger, correctly says at the talk page there "I don't think this [Trigonometric number] concept is really notable..." Or is it? Ponor (talk) 18:22, 26 November 2021 (UTC)

text:   – this seems like an even more artificial concept then. Personally I’d then recommend describing these "trigonometric numbers" in a footnote on some other page, at most. Doesn’t seem notable enough for its own article. –jacobolus (t) 00:15, 27 November 2021 (UTC)
replies: text:  Right, it’s not really the concept of “trigonometric number” that’s notable, it’s everything else —- the connections to roots of unity and constructibility (and cyclotomic integers), the basic square root expressions and the methods for generating them. Danstronger (talk) 03:43, 27 November 2021 (UTC)

text: I don't have a problem with "exact", as in exact trigonometric values or exact values of trigonometric functions. We ought to consider the potential readership: trigonometry is a secondary-school subject, and saying "exact value" will be more meaningful to that audience than anything involving "cyclotomic" or even "algebraic". See, for example, this precalculus textbook, or this one, or the Schaum's study guide for trigonometry. While the result of taking the sine or cosine of any number is tautologically "exact", as noted above, I think that in this case, fussing over that is pedantry at the expense of clarity. By contrast, "trigonometric number" is much less commonplace. XOR'easter (talk) 18:36, 27 November 2021 (UTC)
replies: text: I think the content of this article seems to belong to wikibooks rather than wikipedia. --SilverMatsu (talk) 02:42, 4 December 2021 (UTC)
thread_title:  Very selective merge to [[Trigonometric number]]?