A019827 Decimal expansion of sin(Pi/10) (angle of 18 degrees).

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, 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, 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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy
• changed

Comments

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

Also the imaginary part of i^(1/5). - Stanislav Sykora, Apr 25 2012

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

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

Quadratic number of denominator 2 and minimal polynomial 4x^2 + 2x - 1. - Charles R Greathouse IV, May 13 2019

Largest superstable width of the logistic map (see Finch). - Stefano Spezia, Nov 23 2024

Examples

			0.30901699437494742410229341718281905886015458990288143106772431135263...
		

References

• 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.

Links

• Zak Seidov, Table of n, a(n) for n = 0..999
• Hideyuki Ohtsuka, Problem B-1237, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 56, No. 4 (2018), p. 366; A Telescoping Product, Solution to Problem B-1237 by Steve Edwards, ibid., Vol. 57, No. 4 (2019), pp. 369-370.
• Wikipedia, Exact trigonometric constants.
• Index entries for algebraic numbers, degree 2.

Crossrefs

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

Programs

Mathematica

RealDigits[Sin[18 Degree], 10, 108][[1]] (* Alonso del Arte, Apr 20 2015 *)

PARI

sin(Pi/10) \\ Charles R Greathouse IV, Feb 03 2015

PARI

polrootsreal(4*x^2 + 2*x - 1)[2] \\ Charles R Greathouse IV, Feb 03 2015

Formula

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

Equals 1/(1 + sqrt(5)). - Omar E. Pol, Nov 15 2007

Equals 1/A134945. - R. J. Mathar, Jan 17 2021

Equals 2*A019818*A019890. - R. J. Mathar, Jan 17 2021

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

Equals Product_{k>=1} (1 - 1/A055588(k)). - Amiram Eldar, Nov 28 2024

Equals A094214/2 = 1-A187798 = A341332/Pi = (A377697-2)/3. - Hugo Pfoertner, Nov 28 2024

This^2 + A019881^2 = 1. - R. J. Mathar, Aug 31 2025