A019881 Decimal expansion of sin(2*Pi/5) (sine of 72 degrees).

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, 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, 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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy

Comments

Circumradius of pentagonal pyramid (Johnson solid 2) with edge 1. - Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Circumscribed sphere radius for a regular icosahedron with unit edges. - Stanislav Sykora, Feb 10 2014

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

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

This is length ratio of one half of any diagonal in the regular pentagon and the circumscribing radius. - Wolfdieter Lang, Jan 07 2018

Quartic number of denominator 2 and minimal polynomial 16x^4 - 20x^2 + 5. - Charles R Greathouse IV, May 13 2019

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

Examples

			0.95105651629515357211643933337938214340569863412575022244730564443015317008...
		

References

• Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Golden Rhombus.
• Wikipedia, Exact trigonometric constants.
• Wikipedia, Platonic solid.
• Wolfram Alpha, Johnson solid 2.
• Index entries for algebraic numbers, degree 4.

Crossrefs

Cf. A001622, A019827, A102208, A179290 (inverse), A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A182007.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A179296 (dodecahedron), A187110 (tetrahedron). - Stanislav Sykora, Feb 10 2014

Programs

Magma

SetDefaultRealField(RealField(100)); Sqrt((5 + Sqrt(5))/8); // G. C. Greubel, Nov 02 2018

Maple

Digits:=100: evalf(sin(2*Pi/5)); # Wesley Ivan Hurt, Sep 01 2014

Mathematica

RealDigits[Sqrt[(5 + Sqrt[5])/8], 10, 111]  (* Robert G. Wilson v *)
RealDigits[Sin[2 Pi/5], 10, 111][[1]] (* Robert G. Wilson v, Jan 07 2018 *)

PARI

default(realprecision, 120);
real(I^(1/5)) \\ Rick L. Shepherd, Apr 10 2017

Formula

Equals sqrt((5+sqrt(5))/8) = cos(Pi/10). - Zak Seidov, Nov 18 2006

Equals 2F1(13/20,7/20;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008

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

Equals A001622*A182007/2. - Stanislav Sykora, Feb 10 2014

Equals sin(2*Pi/5) = sqrt(2 + phi)/2 = -sin(3*Pi/5), with phi = A001622 - Wolfdieter Lang, Jan 07 2018

Equals 2*A019845*A019863. - R. J. Mathar, Jan 17 2021