A019829 -id:A019829 - cp's OEIS frontend

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

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

A019889 Decimal expansion of sine of 80 degrees = cos(Pi/18).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.9848077530122080593667430245895230136706432517198424187900...

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

Cf. A019849, A019859.

Digits:=100: evalf(sin(4*Pi/9)); # Wesley Ivan Hurt, Sep 01 2014

RealDigits[ Cos[Pi/18], 10, 111] (* Robert G. Wilson v *)
RealDigits[Sin[80 Degree],10,120][[1]] (* Harvey P. Dale, Apr 16 2022 *)

real(I^(1/9)) \\ Michel Marcus, Nov 29 2013

Equals 2F1(7/12,5/12;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
Also the real part of I^(1/9). - Stanislav Sykora, Nov 29 2013
Equals sin(4*Pi/9). - Wesley Ivan Hurt, Sep 01 2014
Equals 2*A019849*A019859. - R. J. Mathar, Jan 17 2021
Largest positive root of 64*x^6 - 96*x^4 + 36*x^2 - 3. - Artur Jasinski, May 09 2025
Other roots are +- A019849 and +- A019829. - R. J. Mathar, Aug 29 2025
4*this^3 -3*this = A010527. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/12,1/12; 1/2 ; 3/4). - R. J. Mathar, Aug 31 2025

A272488 Decimal expansion of the edge length of a regular 9-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 01 2016

The edge length e(m) of a regular m-gon is e(m) = 2*sin(Pi/m). In this case, m = 9, and the constant, a = e(9), is not constructible using a compass and a straightedge (see A004169). With an odd m, in fact, e(m) would be constructible only if m were a Fermat prime (A019434).

			0.6840402866513374660881992293645191615261667350283212569300969953...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 6

Cf. A004169, A019434.

Edge lengths of nonconstructible n-gons: A272487 (n=7), A272489 (n=11), A272490 (n=13), A255241 (n=14), A130880 (n=18), A272491 (n=19).

RealDigits[N[2Sin[Pi/9], 100]][[1]] (* Robert Price, May 01 2016 *)

PARI
```
2*sin(Pi/9)
```

Equals 2*sin(Pi/9) = 2*cos(Pi*7/18) = 2*A019829.
Equals Im((4+4*sqrt(3)*i)^(1/3)). - Gerry Martens, Mar 19 2024
A root of x^6 -6*x^4 +9*x^2 -3 =0. - R. J. Mathar, Aug 29 2025

A019849 Decimal expansion of sine of 40 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This sequence is also the decimal expansion of cosine of 50 degrees. - Mohammad K. Azarian, Jun 29 2013

A sextic number with denominator 2. - Charles R Greathouse IV, Nov 05 2017

			0.642787609...

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

RealDigits[ Sin[ 2Pi/9], 10, 111][[1]] (* Robert G. Wilson v, Feb 16 2011 *)
RealDigits[Sin[40 Degree],10,120][[1]] (* Harvey P. Dale, Mar 27 2015 *)

sin(2/9*Pi) \\ Charles R Greathouse IV, Nov 05 2017

polrootsreal(64*x^6-96*x^4+36*x^2-3)[5] \\ Charles R Greathouse IV, Feb 05 2025

Equals cos(5*Pi/18) = 2F1(11/12,1/12;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
Equals 2*A019829*A019879. - R. J. Mathar, Jan 17 2021
A root of 64*x^6 - 96*x^4 + 36*x^2 - 3. - R. J. Mathar, Aug 29 2025

A019918 Decimal expansion of tangent of 20 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the decimal expansion of cotangent of 70 degrees. - Mohammad K. Azarian, Jun 30 2013

			0.36397023426620236135104788277683404389047...

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

Cf. A019938 (tan(2*Pi/9)).

evalf(tan(Pi/9), 100); # Bernard Schott, Apr 19 2022

First[RealDigits[Tan[Pi/9], 10, 100]] (* Paolo Xausa, Aug 01 2024 *)

tan(Pi/9) \\ Charles R Greathouse IV, Feb 05 2025

polrootsreal(x^6-33*x^4+27*x^2-3)[4] \\ Charles R Greathouse IV, Feb 05 2025

Equals tan(Pi/9) = A019829/A019879. - Bernard Schott, Apr 19 2022

Smallest positive of the 6 real roots of x^6-33*x^4+27*x^2-3=0. (Other A019978, A019938). - R. J. Mathar, Aug 31 2025

A019968 Decimal expansion of tangent of 70 degrees.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Also the decimal expansion of cotangent of 20 degrees. - Ivan Panchenko, Sep 01 2014

An algebraic number of degree 6 and denominator 3. - Charles R Greathouse IV, Aug 27 2017

			2.7474774194546222787616640264976727177518725991708258215...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Index entries for algebraic numbers, degree 6

Cf. A019879 (sine of 70 degrees).

SetDefaultRealField(RealField(100)); R:= RealField(); Tan(7*Pi(R)/18); // G. C. Greubel, Nov 21 2018

RealDigits[Tan[7*Pi/18], 10, 100][[1]] (* G. C. Greubel, Nov 21 2018 *)

tan(7*Pi/18) \\ Charles R Greathouse IV, Aug 27 2017

numerical_approx(tan(7*pi/18), digits=100) # G. C. Greubel, Nov 21 2018

Equals A019879/A019829. - R. J. Mathar, Aug 29 2025

Largest positive of the 6 real-values roots of 3*x^6 -27*x^4 +33*x^2 -1 =0. (Others: A019948, A019908). - R. J. Mathar, Aug 29 2025

A121602 Decimal expansion of cosecant of 20 degrees = csc(Pi/9).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Aug 09 2006

1 + csc(Pi/9) is the radius of the smallest circle into which 11 unit circles can be packed ("r=3.923+ Proved by Melissen in 1994.", according to the Friedman link, which has a diagram). csc(Pi/9) [=1/A019829] is the distance between the center of the larger circle and the center of each unit circle that touches the larger circle.

			2.9238044001630872522327544133662917...

Index entries for algebraic numbers, degree 6.

Cf. A019829 (1/A121602), A121601.

RealDigits[Csc[20 Degree],10,120][[1]] (* Harvey P. Dale, May 27 2023 *)

PARI
```
1/sin(Pi/9)
```

polrootsreal(3*x^6-36*x^4+96*x^2-64)[6] \\ Charles R Greathouse IV, Feb 04 2025

A323601 Decimal expansion of sin(Pi/7).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jan 19 2019

			0.43388373911755812047576833284835875460999072778745987644454730353220325...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for algebraic numbers, degree 6.

Cf. A019829 (sin(Pi/9)), A232736 (sin(Pi/14)).

Cf. A121598, A272487.

SetDefaultRealField(RealField(100)); R:= RealField(); Sin(Pi(R)/7); // G. C. Greubel, Feb 08 2019

Mathematica
```
RealDigits[Sin[Pi/7], 10, 120][[1]]
```

default(realprecision, 100); sin(Pi/7) \\ G. C. Greubel, Feb 08 2019

polrootsreal(64*x^6-112*x^4+56*x^2-7)[4] \\ Charles R Greathouse IV, Feb 05 2025

numerical_approx(sin(pi/7), digits=100) # G. C. Greubel, Feb 08 2019

Root of the equation 64*x^6 - 112*x^4 + 56*x^2 - 7 = 0. (Other +- A232735 and +- 0.7818314... = +- cos(3*Pi/14))
Equals sqrt((196 + 7*i*2^(2/3)*(21*i*sqrt(3) - 7)^(1/3)*(i + sqrt(3)) + i*2^(4/3)*(21*i*sqrt(3) - 7)^(2/3)*(2*i + sqrt(3)))/336), where i is the imaginary unit.
Equals cos(5*Pi/14).
From Gleb Koloskov, Jul 15 2021: (Start)
Positive root of the equation x^3 + sqrt(7)/2*x^2 - sqrt(7)/8 = 0.
Equals ((4*sqrt(7)*(13+3*sqrt(3)*i))^(1/3)+28*(4*sqrt(7)*(13+3*sqrt(3)*i))^(-1/3)-2*sqrt(7))/12, where i is the imaginary unit. (End)
Equals 1/A121598 = A272487/2. - Hugo Pfoertner, Dec 15 2024
This^2 + A073052^2=1. - R. J. Mathar, Aug 31 2025

A019886 Decimal expansion of sine of 77 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Equals sin(77*Pi/180). - Wesley Ivan Hurt, Sep 01 2014

An algebraic number of degree 48 and denominator 2. - Charles R Greathouse IV, Nov 06 2017

			0.974370064785235228539694480088268833005120988944567944597972222...

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

Digits:=100: evalf(sin(77*Pi/180)); # Wesley Ivan Hurt, Sep 01 2014

RealDigits[Sin[77 Degree],10,120][[1]] (* Harvey P. Dale, Jan 25 2013 *)

sin(77/180*Pi) \\ Charles R Greathouse IV, Nov 06 2017

Equals A019875 * A019888 + A019820 * A019833 = A019879 * A019892 + A019816 * A019829. - R. J. Mathar, Jan 27 2021

A019834 Decimal expansion of sine of 25 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 12 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 12

RealDigits[Sin[25 Degree],10,120][[1]] (* Harvey P. Dale, Mar 18 2018 *)

sin(5*Pi/36) \\ Charles R Greathouse IV, Aug 27 2017

Equals A019829 * A019894 + A019814 * A019879. - R. J. Mathar, Jan 27 2021

cp's OEIS Frontend

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

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A019889 Decimal expansion of sine of 80 degrees = cos(Pi/18).

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A272488 Decimal expansion of the edge length of a regular 9-gon with unit circumradius.

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A019849 Decimal expansion of sine of 40 degrees.

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A019918 Decimal expansion of tangent of 20 degrees.

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Maple

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A019968 Decimal expansion of tangent of 70 degrees.

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A121602 Decimal expansion of cosecant of 20 degrees = csc(Pi/9).

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