A019872 -id:A019872 - cp's OEIS frontend

A019863 Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).

8, 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

N. J. A. Sloane

Midsphere radius of regular icosahedron with unit edges.
Also half of the golden ratio (A001622). - Stanislav Sykora, Jan 30 2014
Andris Ambainis (see Aaronson link) observes that combining the results of Barak-Hardt-Haviv-Rao with Dinur-Steurer yields the maximal probability of winning n parallel repetitions of a classical CHSH game (see A201488) asymptotic to this constant to the power of n, an improvement on the naive probability of (3/4)^n. (All the random bits are received upfront but the players cannot communicate or share an entangled state.) - Charles R Greathouse IV, May 15 2014
This is the height h of the isosceles triangle in a regular pentagon, in length units of the circumscribing radius, formed by a side as base and two adjacent radii. h = sin(3*Pi/10) = cos(Pi/5) (radius 1 unit). - Wolfdieter Lang, Jan 08 2018
Also the limiting value(L) of "r" which is abscissa of the vertex of the parabola  F(n)*x^2 - F(n+1)*x + F(n + 2)(where F(n)=A000045(n) are the Fibonacci numbers and n>0). - Burak Muslu, Feb 24 2021

			0.80901699437494742410229341718281905886015458990288143106772431135263...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Scott Aaronson, The NEW Ten Most Annoying Questions in Quantum Computing (2014).
Boaz Barak, Moritz Hardt, Ishay Haviv, and Anup Rao, Rounding Parallel Repetitions of Unique Games (2008).
Irit Dinur and David Steurer, Analytical approach to parallel repetition, arXiv:1305.1979 [cs.CC], 2013-2014.
Michael Penn, a golden value of cosine., YouTube video, 2021.
Wikipedia, Exact trigonometric constants.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A001611, A001622, A019827, A134972, A134944, A134946, A187426, A296182.

Platonic solids midradii: A020765 (tetrahedron), A020761 (octahedron), A010503 (cube), A239798 (dodecahedron).

convert(sin(3*Pi/10),radical); # W. Edwin Clark, May 24 2023
Digits:=100; evalf((1+sqrt(5))/4); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[(1 + Sqrt[5])/4, 10, 111] (* Robert G. Wilson v *)
RealDigits[Sin[54 Degree],10,120][[1]] (* Harvey P. Dale, Apr 21 2018 *)

(1+sqrt(5))/4 \\ Charles R Greathouse IV, Jan 16 2012

Equals (1+sqrt(5))/4 = cos(Pi/5) = sin(3*Pi/10). - R. J. Mathar, Jun 18 2006
Equals 2F1(4/5,1/5;1/2;3/4) / 2 = A019827 + 1/2. - R. J. Mathar, Oct 27 2008
Equals A001622 / 2. - Stanislav Sykora, Jan 30 2014
phi / 2 = (i^(2/5) + i^(-2/5)) / 2 = i^(2/5) - (sin(Pi/5))*i = i^(-2/5) + (sin(Pi/5))*i = i^(2/5) - (cos(3*Pi/10))*i = i^(-2/5) + (cos(3*Pi/10))*i. - Jaroslav Krizek, Feb 03 2014
Equals 1/A134972. - R. J. Mathar, Jan 17 2021
Equals 2*A019836*A019872. - R. J. Mathar, Jan 17 2021
Equals (A094214 + 1)/2 or 1/(2*A094214). - Burak Muslu, Feb 24 2021
Equals hypergeom([-2/5, -3/5], [6/5], -1) = hypergeom([-1/5, 3/5], [6/5], 1) = hypergeom([1/5, -3/5], [4/5], 1). - Peter Bala, Mar 04 2022
Equals Product_{k>=1} (1 - (-1)^k/A001611(k)). - Amiram Eldar, Nov 28 2024
Equals 2*A134944 = 3*A134946 = A187426-11/10 = A296182-1. - Hugo Pfoertner, Nov 28 2024
Equals A134945/4. Root of 4*x^2-2*x-1=0. - R. J. Mathar, Aug 29 2025

A019890 Decimal expansion of sine of 81 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the real part of i^(1/10). - Stanislav Sykora, Apr 25 2012
Equals sin(9*Pi/20). - Wesley Ivan Hurt, Sep 01 2014
An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

			0.98768834059513772619004024769343726075840686158988043492390480163...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 8

Digits:=100: evalf(sin(9*Pi/20)); # Wesley Ivan Hurt, Sep 01 2014

RealDigits[Sin[81 Degree],10,120][[1]] (* Harvey P. Dale, Aug 11 2012 *)

sin(9*Pi/20) \\ Charles R Greathouse IV, Aug 27 2017

Equals cos(Pi/20) = sqrt((1+A019881)/2) = sqrt(1-A019818^2) = sqrt(5-sqrt(5))*(sqrt(5)+sqrt(5+2*sqrt(5)))/(4*sqrt(5)). - R. J. Mathar, Jun 18 2006
Equals A019863 * A019872 + A019836 * A019845 = A019887 * A019896 + A019812 * A019821. - R. J. Mathar, Jan 27 2021
Root of 256*x^8 -512*x^6 +304*x^4 -48*x^2+1=0. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/10,1/10;1/2;1/2). - R. J. Mathar, Aug 31 2025

A019925 Decimal expansion of tangent of 27 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the decimal expansion of cotangent of 63 degrees. - Mohammad K. Azarian, Jun 30 2013

			tan(3*Pi/20) = 0.509525449...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 8

Root of x^8-44*x^6+166*x^4-44*x^2+1=0. - R. J. Mathar, Aug 31 2025

Equals A019836 / A019872. - R. J. Mathar, Aug 31 2025

A019961 Decimal expansion of tangent of 63 degrees.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Also the decimal expansion of cotangent of 27 degrees. - Ivan Panchenko, Sep 01 2014

			1.96261050550515058230464042621189498505671...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 8

Cf. A019872 (sine of 63 degrees).

SetDefaultRealField(RealField(100)); R:= RealField(); Tan(7*Pi(R)/20); // G. C. Greubel, Nov 21 2018

RealDigits[Tan[7*Pi/20], 10, 100][[1]] (* G. C. Greubel, Nov 21 2018 *)

default(realprecision, 100); tan(7*Pi/20) \\ G. C. Greubel, Nov 21 2018

numerical_approx(tan(7*pi/20), digits=100) # G. C. Greubel, Nov 21 2018

Equals cot(3*Pi/20) = sqrt(5) - 1 + sqrt(5 - 2*sqrt(5)). - G. C. Greubel, Nov 21 2018

Root of x^8-44*x^6+166*x^4-44*x^2+1=0. - R. J. Mathar, Aug 31 2025

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A019863 Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).

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A019890 Decimal expansion of sine of 81 degrees.

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A019925 Decimal expansion of tangent of 27 degrees.

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A019961 Decimal expansion of tangent of 63 degrees.

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