A019889 -id:A019889 - cp's OEIS frontend

A019819 Decimal expansion of sine of 10 degrees.

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

Offset: 0

N. J. A. Sloane

Also the imaginary part of i^(1/9). - Stanislav Sykora, Apr 25 2012

			0.173648177...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Vladimir Ivanovich Smirnov, A course of higher mathematics, vol. 1 , Pergamon Press, 1964, p. 342.
Index entries for algebraic numbers, degree 3.

Cf. A019814.

First[RealDigits[Root[1 - 6 #1 + 8 #1^3 &, 2], 10, 100]] (* Artur Jasinski, Oct 28 2008 *)
RealDigits[ Sin[Pi/18], 10, 111]  (* Robert G. Wilson v *)

sin(Pi/18) \\ Charles R Greathouse IV, Apr 25 2012

Equals cos(4*Pi/9) = 2F1(7/6,-1/6;1/2;3/4) / 2 = - 2F1(4/3,-1/3;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
From Artur Jasinski, Oct 28 2008: (Start)
Decimal expansion of root of cubic polynomial 1 - 6*x + 8*x^3. (Others A019859, -A019879)
Decimal expansion of casus irreducibilis:
(1/2) * (((-i*sqrt(3) - 1)/2)^(2/3) + ((i*sqrt(3) - 1)/2)^(2/3)). (End)
Equals 2 * A019814 * A019894. - R. J. Mathar, Jan 17 2021
This^2 + A019889^2 = 1. - R. J. Mathar, Aug 31 2025

A130880 Decimal expansion of 2*sin(Pi/18).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 26 2007

Also: a bond percolation threshold probability on the triangular lattice.

Also: the edge length of a regular 18-gon with unit circumradius. Such an m-gon is not constructible using a compass and a straightedge (see A004169). With an even m, in fact, it would be constructible only if the (m/2)-gon were constructible, which is not true in this case (see A272488). - Stanislav Sykora, May 01 2016

			0.347296355333860697703433253538629592...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.18.1, p. 373.

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Steven R. Finch, Several Constants Arising in Statistical Mechanics, Annals Combinat., Vol. 3, Issue 2-4 (1999), pp. 323-335.
Wikipedia, Constructible number.
Wikipedia, Percolation threshold.
Wikipedia, Regular polygon.
Index entries for algebraic numbers, degree 3.

Cf. A004169, A019819, A019829, A019889, A063289, A133749, A178959, A332437, A332438.

Edge lengths of nonconstructible n-gons: A272487 (n=7), A272488 (n=9), A272489 (n=11), A272490 (n=13), A255241 (n=14), A272491 (n=19). - Stanislav Sykora, May 01 2016

RealDigits[N[2Sin[Pi/18], 100]][[1]] (* Robert Price, May 01 2016 *)

PARI
```
2*sin(Pi/18)
```

Equals 2*A019819 = A019829/A019889.
Algebraic number with minimal polynomial over Q equal to x^3 - 3*x + 1, a cyclic cubic, having zeros 2*sin(Pi/18) (= 2*cos(4*Pi/9)), 2*sin(5*Pi/18) (= 2*cos(2*Pi/9)) and -2*sin(7*Pi/18) (= -2*cos(Pi/9)). Cf. A332437. - Peter Bala, Oct 23 2021
Equals 2 + rho(9) - rho(9)^2, an element of the extension field Q(rho(9)), with rho(9) = 2*cos(Pi/9) = A332437 with minimal polynomial x^3 - 3*x - 1 over Q. - Wolfdieter Lang, Sep 20 2022
Equals -1 + Product_{k>=3} (1 - (-1)^k/A063289(k)). - Amiram Eldar, Nov 22 2024
Equals A133749/2 = 1 - A178959. - Hugo Pfoertner, Dec 15 2024

A019824 Decimal expansion of sine of 15 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the imaginary part of i^(1/6). - Stanislav Sykora, Apr 25 2012

			0.258819045102520762348898837624048328349068901319930513814003207315...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mark B. Villarino, Legendre's Singular Modulus, arXiv:2002.07250 [math.HO], 2020.
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 4

Cf. A002194, A020765, A101263.

RealDigits[ Sin[ Pi/12], 10, 111][[1]] (* Or *) RealDigits[(Sqrt[3] - 1)/(2 Sqrt[2]), 10, 111][[1]] (* Robert G. Wilson v *)
RealDigits[Sin[15 Degree],10,120][[1]] (* Harvey P. Dale, Jul 16 2016 *)

sin(Pi/12) \\ Charles R Greathouse IV, Apr 25 2012

Equals (sqrt(3)-1)/(2*sqrt(2)) = (A002194 -1) * A020765 = sin(Pi/12). - R. J. Mathar, Jun 18 2006
Equals 2F1(9/8,-1/8;1/2;3/4) / 2 = - 2F1(11/8,-3/8;1/2;3/4) / 2 = cos(5*Pi/12). - R. J. Mathar, Oct 27 2008
Equals sqrt(2 - sqrt(3))/2 = (1/2) * A101263. - Amiram Eldar, Aug 05 2020
Equals A019819 * A019894 + A019814 * A019889 = A019821 * A019896 + A019812 * A019887. - R. J. Mathar, Jan 27 2021
This^2 + A019884^2=1. - R. J. Mathar, Aug 31 2025
Smallest positive of the 4 real-valued roots of 16*x^4-16*x^2+1=0. - R. J. Mathar, Aug 31 2025

A019829 Decimal expansion of sine of 20 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.34202014332566873304409961468225958076308336751416062846504849768471476...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 6

Cf. A323601.

RealDigits[ Sin[Pi/9], 10, 111][[1]]  (* Robert G. Wilson v *)

/* for x = 20 degrees, sin(9x) = 0 */
/* so sin(x) is a zero of this polynomial */
sin_9(x)=9*x-120*x^3+432*x^5-576*x^7+256*x^9
x=34;y=100;print(3);print(4);
for(digits=1, 110, {d=0;y=y*10;while(sin_9((10*x+d)/y) > 0, d++);
d--; /* while loop overshoots correct digit */
print(d); x=10*x+d})
\\ Michael B. Porter, Jan 27 2010

sin(Pi/9) \\ Charles R Greathouse IV, Feb 04 2025

Equals cos(7*Pi/18) = 2F1(13/12,-1/12;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
Root of the equation 64*x^6 - 96*x^4 + 36*x^2 - 3 = 0. - Vaclav Kotesovec, Jan 19 2019 (other A019849, A019889)
Equals sqrt(8 - 2^(4/3)*(1 + i*sqrt(3))^(2/3) + i*2^(2/3)*(1 + i*sqrt(3))^(1/3)*(i + sqrt(3)))/4, where i is the imaginary unit. - Vaclav Kotesovec, Jan 19 2019
Equals 2*A019819 *A019889. - R. J. Mathar, Jan 17 2021
This^2 + A019879^2=1. - R. J. Mathar, Aug 31 2025

A019814 Decimal expansion of sine of 5 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 12 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

			0.087155...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 12

Cf. A019810, A019813, A019895, A019898.

Join[{0},RealDigits[Sin[5 Degree],10,120][[1]]] (* Harvey P. Dale, Jun 06 2018 *)

sin(Pi/36) \\ Charles R Greathouse IV, Aug 27 2017

Equals A019813 * A019898 + A019810 * A019895. - R. J. Mathar, Jan 27 2021
Equals sin(Pi/36) = cos(17*Pi/36). 2*this^2-1 = -A019889. - R. J. Mathar, Aug 29 2025
One of the 12 real-valued roots of 4096*x^12 -12288*x^10 +13824*x^8 -7168*x^6 +1680*x^4 -144*x^2 +1 =0. - R. J. Mathar, Aug 29 2025

Zero added in front by R. J. Mathar, Feb 05 2009

A019894 Decimal expansion of sine of 85 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 12

Cf. A019814, A019819, A019826, A019831, A019877, A019882, A019889, A019894.

RealDigits[Sin[17*Pi/36],10,99][[1]] (* Stefano Spezia, Feb 09 2025 *)

Equals cos(Pi/36) = 2F1(13/24,11/24;1/2;3/4) / 2 . - R. J. Mathar, Oct 27 2008

Equals A019877 * A019882 + A019826 * A019831 = A019889 * A019894 + A019814 * A019819. - R. J. Mathar, Jan 27 2021

A232735 Decimal expansion of the real part of I^(1/7), or cos(Pi/14).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding imaginary part is in A232736.

Root of the equation -7 + 56*x^2 - 112*x^4 + 64*x^6 = 0. - Vaclav Kotesovec, Apr 04 2021

			0.974927912181823607018131682993931217232785800619997437648...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
A. Arman, A. Bondarenko, and A. Prymak, Convex bodies of constant width with exponential illumination number, arXiv:2304.10418 [math.MG], 2023.
Index entries for algebraic numbers, degree 6

Cf. A232736 (imaginary part), A010503 (real(I^(1/2))), A010527 (real(I^(1/3))), A144981 (real(I^(1/4))), A019881 (real(I^(1/5))), A019884 (real(I^(1/6))), A232737 (real(I^(1/8))), A019889 (real(I^(1/9))), A019890 (real(I^(1/10))).

R:= RealField(100); Cos(Pi(R)/14); // G. C. Greubel, Sep 19 2022

RealDigits[Cos[Pi/14],10,120][[1]] (* Harvey P. Dale, Dec 15 2018 *)

numerical_approx(cos(pi/14), digits=120) # G. C. Greubel, Sep 19 2022

2*this^2 -1 = A073052. - R. J. Mathar, Aug 29 2025

Equals 2F1(-1/14,1/14;1/2;1) . - R. J. Mathar, Aug 31 2025

A019820 Decimal expansion of sine of 11 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 48 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 48.

Cf. A019810, A019819, A019889, A019898.

RealDigits[Sin[11*Pi/180],10,99][[1]] (* Stefano Spezia, Feb 09 2025 *)

sin(11*Pi/180) \\ Charles R Greathouse IV, Aug 27 2017

Equals A019819 * A019898 + A019810 * A019889. - R. J. Mathar, Jan 27 2021

A019908 Decimal expansion of tangent of 10 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the decimal expansion of cotangent of 80 degrees. - Mohammad K. Azarian, Jun 30 2013

			0.176326980708464973471090386868618986121633...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 6.

Cf. A019918.

RealDigits[Tan[10 Degree],10,120][[1]] (* Harvey P. Dale, Oct 14 2012 *)

tan(Pi/18) \\ Hugo Pfoertner, Aug 04 2024

A root of 3*x^6 -27*x^4 +33*x^2 -1 =0 (others A019968, A019948). - R. J. Mathar, Aug 29 2025

tan(Pi/18) = A019819/A019889. - R. J. Mathar, Aug 31 2025

A232737 Decimal expansion of the real part of I^(1/8), or cos(Pi/16).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding imaginary part is in A232738.

			0.9807852804032304491261822361342390369739337308933360950029160885453...

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

Cf. A232738 (imaginary part), A010503 (real(I^(1/2))), A010527 (real(I^(1/3))), A144981 (real(I^(1/4))), A019881 (real(I^(1/5))), A019884 (real(I^(1/6))), A232735 (real(I^(1/7))), A019889 (real(I^(1/9))), A019890 (real(I^(1/10))).

RealDigits[Cos[Pi/16], 10, 120][[1]] (* Amiram Eldar, Jun 29 2023 *)

real(I^(1/8)) \\ Seiichi Manyama, Apr 04 2021

cos(Pi/16) \\ Seiichi Manyama, Apr 04 2021

sqrt(2+sqrt(2+sqrt(2)))/2 \\ Seiichi Manyama, Apr 04 2021

Equals (1/2) * sqrt(2+sqrt(2+sqrt(2))). - Seiichi Manyama, Apr 04 2021
Root of 128*x^8 -256*x^6 +160*x^4 -32*x^2 +1 = 0. - R. J. Mathar, Aug 29 2025
2*this^2 -1 = A144981. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/8,1/8;1/2;1/2). - R. J. Mathar, Aug 31 2025

cp's OEIS Frontend

A019819 Decimal expansion of sine of 10 degrees.

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A130880 Decimal expansion of 2*sin(Pi/18).

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A019824 Decimal expansion of sine of 15 degrees.

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A019829 Decimal expansion of sine of 20 degrees.

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A019814 Decimal expansion of sine of 5 degrees.

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A019894 Decimal expansion of sine of 85 degrees.

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A232735 Decimal expansion of the real part of I^(1/7), or cos(Pi/14).

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