%I #85 Jan 05 2025 19:51:34

%S 3,0,9,0,1,6,9,9,4,3,7,4,9,4,7,4,2,4,1,0,2,2,9,3,4,1,7,1,8,2,8,1,9,0,

%T 5,8,8,6,0,1,5,4,5,8,9,9,0,2,8,8,1,4,3,1,0,6,7,7,2,4,3,1,1,3,5,2,6,3,

%U 0,2,3,1,4,0,9,4,5,1,2,2,4,8,5,3,6,0,3,6,0,2,0,9,4,6,9,5,5,6,8

%N Decimal expansion of sin(Pi/10) (angle of 18 degrees).

%C Decimal expansion of cos(2*Pi/5) (angle of 72 degrees).

%C Also the imaginary part of i^(1/5). - _Stanislav Sykora_, Apr 25 2012

%C One of the two roots of 4x^2 + 2x - 1 (the other is the sine of 54 degrees times -1). - _Alonso del Arte_, Apr 25 2015

%C This is the height h of the isosceles triangle in a regular pentagon inscribed in a unit circle, formed by a diagonal as base and two adjacent radii. h = cos(2*Pi/5) = sin(Pi/10). - _Wolfdieter Lang_, Jan 08 2018

%C Quadratic number of denominator 2 and minimal polynomial 4x^2 + 2x - 1. - _Charles R Greathouse IV_, May 13 2019

%C Largest superstable width of the logistic map (see Finch). - _Stefano Spezia_, Nov 23 2024

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.9 and 8.19, pp. 66, 535.

%H Zak Seidov, <a href="/A019827/b019827.txt">Table of n, a(n) for n = 0..999</a>

%H Hideyuki Ohtsuka, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbSolnNov2018.pdf">Problem B-1237</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 56, No. 4 (2018), p. 366; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/ElemProbSolnNov2019.pdf">A Telescoping Product</a>, Solution to Problem B-1237 by Steve Edwards, ibid., Vol. 57, No. 4 (2019), pp. 369-370.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Exact_trigonometric_constants">Exact trigonometric constants</a>.

%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.

%F Equals (sqrt(5) - 1)/4 = (phi - 1)/2 = 1/(2*phi), with phi from A001622.

%F Equals 1/(1 + sqrt(5)). - _Omar E. Pol_, Nov 15 2007

%F Equals 1/A134945. - _R. J. Mathar_, Jan 17 2021

%F Equals 2*A019818*A019890. - _R. J. Mathar_, Jan 17 2021

%F Equals Product_{k>=1} 1 - 1/(phi + phi^k), where phi is the golden ratio (A001622) (Ohtsuka, 2018). - _Amiram Eldar_, Dec 02 2021

%F Equals Product_{k>=1} (1 - 1/A055588(k)). - _Amiram Eldar_, Nov 28 2024

%F Equals A094214/2 = 1-A187798 = A341332/Pi = (A377697-2)/3. - _Hugo Pfoertner_, Nov 28 2024

%e 0.30901699437494742410229341718281905886015458990288143106772431135263...

%t RealDigits[Sin[18 Degree], 10, 108][[1]] (* _Alonso del Arte_, Apr 20 2015 *)

%o (PARI) sin(Pi/10) \\ _Charles R Greathouse IV_, Feb 03 2015

%o (PARI) polrootsreal(4*x^2 + 2*x - 1)[2] \\ _Charles R Greathouse IV_, Feb 03 2015

%Y Cf. A001622, A019845, A019863, A055588, A094214, A134945, A187798, A341332, A377697.

%K nonn,cons,easy

%O 0,1

%A _N. J. A. Sloane_