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%I #74 Aug 21 2023 11:37:07
%S 9,5,1,0,5,6,5,1,6,2,9,5,1,5,3,5,7,2,1,1,6,4,3,9,3,3,3,3,7,9,3,8,2,1,
%T 4,3,4,0,5,6,9,8,6,3,4,1,2,5,7,5,0,2,2,2,4,4,7,3,0,5,6,4,4,4,3,0,1,5,
%U 3,1,7,0,0,8,5,1,9,3,5,0,1,7,1,8,7,9,2,8,1,0,9,7,0,8,1,1,3,8,1
%N Decimal expansion of sin(2*Pi/5) (sine of 72 degrees).
%C Circumradius of pentagonal pyramid (Johnson solid 2) with edge 1. - _Vladimir Joseph Stephan Orlovsky_, Jul 19 2010
%C Circumscribed sphere radius for a regular icosahedron with unit edges. - _Stanislav Sykora_, Feb 10 2014
%C Side length of the particular golden rhombus with diagonals 1 and phi (A001622); area is phi/2 (A019863). Thus, also the ratio side/(shorter diagonal) for any golden rhombus. Interior angles of a golden rhombus are always A105199 and A137218. - _Rick L. Shepherd_, Apr 10 2017
%C An algebraic number of degree 4; minimal polynomial is 16x^4 - 20x^2 + 5, which has these smaller, other solutions (conjugates): -A019881 < -A019845 < A019845 (sine of 36 degrees). - _Rick L. Shepherd_, Apr 11 2017
%C This is length ratio of one half of any diagonal in the regular pentagon and the circumscribing radius. - _Wolfdieter Lang_, Jan 07 2018
%C Quartic number of denominator 2 and minimal polynomial 16x^4 - 20x^2 + 5. - _Charles R Greathouse IV_, May 13 2019
%C This gives the imaginary part of one of the members of a conjugate pair of roots of x^5 - 1, with real part (-1 + phi)/2 = A019827, where phi = A001622. A member of the other conjugte pair of roots is (-phi + sqrt(3 - phi)*i)/2 = (-A001622 + A182007*i)/2 = -A001622/2 + A019845*i. - _Wolfdieter Lang_, Aug 30 2022
%H Ivan Panchenko, <a href="/A019881/b019881.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldenRhombus.html">Golden Rhombus</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Exact_trigonometric_constants">Exact trigonometric constants</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Platonic solid">Platonic solid</a>
%H Wolfram Alpha, <a href="http://www.wolframalpha.com/input/?i=Johnson+solid+2">Johnson solid 2</a>
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F Equals sqrt((5+sqrt(5))/8) = cos(Pi/10). - _Zak Seidov_, Nov 18 2006
%F Equals 2F1(13/20,7/20;1/2;3/4) / 2. - _R. J. Mathar_, Oct 27 2008
%F Equals the real part of i^(1/5). - _Stanislav Sykora_, Apr 25 2012
%F Equals A001622*A182007/2. - _Stanislav Sykora_, Feb 10 2014
%F Equals sin(2*Pi/5) = sqrt(2 + phi)/2 = sin(3*Pi/5), with phi = A001622 - _Wolfdieter Lang_, Jan 07 2018
%F Equals 2*A019845*A019863. - _R. J. Mathar_, Jan 17 2021
%e 0.95105651629515357211643933337938214340569863412575022244730564443015317008...
%p Digits:=100: evalf(sin(2*Pi/5)); # _Wesley Ivan Hurt_, Sep 01 2014
%t RealDigits[Sqrt[(5 + Sqrt[5])/8], 10, 111] (* _Robert G. Wilson v_ *)
%t RealDigits[Sin[2 Pi/5], 10, 111][[1]] (* _Robert G. Wilson v_, Jan 07 2018 *)
%o (PARI)
%o default(realprecision, 120);
%o real(I^(1/5)) // _Rick L. Shepherd_, Apr 10 2017
%o (Magma) SetDefaultRealField(RealField(100)); Sqrt((5 + Sqrt(5))/8); // _G. C. Greubel_, Nov 02 2018
%Y Cf. A001622, A019827, A102208, A179290 (inverse), A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A182007.
%Y Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A179296 (dodecahedron), A187110 (tetrahedron). - _Stanislav Sykora_, Feb 10 2014
%K nonn,cons
%O 0,1
%A _N. J. A. Sloane_
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