This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A019879 #38 Aug 31 2025 10:04:29 %S A019879 9,3,9,6,9,2,6,2,0,7,8,5,9,0,8,3,8,4,0,5,4,1,0,9,2,7,7,3,2,4,7,3,1,4, %T A019879 6,9,9,3,6,2,0,8,1,3,4,2,6,4,4,6,4,6,3,3,0,9,0,2,8,6,6,6,2,7,7,4,2,2, %U A019879 1,2,1,0,9,9,5,8,8,9,4,5,8,9,4,9,7,4,5,8,8,9,8,3,7,9,4,8,0,6,7 %N A019879 Decimal expansion of sine of 70 degrees. %C A019879 It is well known that the length sin 70° (cos 20°) is not constructible with ruler and compass, since it is a root of the irreducible polynomial 8x^3 - 6x - 1 and 3 fails to divide any power of 2. - _Jean-François Alcover_, Aug 10 2014 [cf. the Maxfield ref.] %C A019879 A cubic number with denominator 2. - _Charles R Greathouse IV_, Aug 27 2017 %C A019879 From _Peter Bala_, Oct 21 2021: (Start) %C A019879 The minimal polynomial of cos(Pi/9) is 8*x^3 - 6*x - 1 with discriminant (2^6)*(3^4), a square: hence the Galois group of the algebraic number field Q(sin(70°) over Q is the cyclic group of order 3. %C A019879 The two other zeros of the minimal polynomial are cos(5*Pi/9) = - A019819 and cos(7*Pi/9) = - A019859. The mapping z -> 1 - 2*z^2 cyclically permutes the three zeros. The inverse permutation is given by the mapping z -> 2*z^2 - z - 1. (End) %D A019879 J. E. Maxfield and M. W. Maxfield, Abstract Algebra and Solution by Radicals, Dover Publications ISBN 0-486-67121-6, (1992), p. 197. %H A019879 Vincenzo Librandi, <a href="/A019879/b019879.txt">Table of n, a(n) for n = 0..1000</a> %H A019879 Michael Penn, <a href="https://www.youtube.com/watch?v=-1lsy6DceLk">Proof that cos(20°) is irrational</a>, YouTube video, 2022. %H A019879 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a> %F A019879 Equals 2*A019844*A019864. - _R. J. Mathar_, Jan 17 2021 %F A019879 Equals cos(Pi/9) = (1/2)*A332437. - _Peter Bala_, Oct 21 2021 %F A019879 Equals 2F1(-1/6,1/6 ; 1/2; 3/4). - _R. J. Mathar_, Aug 31 2025 %e A019879 0.93969262... %t A019879 RealDigits[Sin[70 Degree],10,120][[1]] (* _Harvey P. Dale_, Aug 17 2012 *) %o A019879 (PARI) cos(Pi/9) \\ _Charles R Greathouse IV_, Aug 27 2017 %Y A019879 Cf. A019819, A019844, A019859, A214778, A332437. %K A019879 nonn,cons,changed %O A019879 0,1 %A A019879 _N. J. A. Sloane_