A019859 -id:A019859 - 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

A019879 Decimal expansion of sine of 70 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

It is well known that the length sin 70° (cos 20°) is not constructible with ruler and compass, since it is a root of the irreducible polynomial 8x^3 - 6x - 1 and 3 fails to divide any power of 2. - Jean-François Alcover, Aug 10 2014 [cf. the Maxfield ref.]
A cubic number with denominator 2. - Charles R Greathouse IV, Aug 27 2017
From Peter Bala, Oct 21 2021: (Start)
The minimal polynomial of cos(Pi/9) is 8*x^3 - 6*x - 1 with discriminant (2^6)*(3^4), a square: hence the Galois group of the algebraic number field Q(sin(70°) over Q is the cyclic group of order 3.
The two other zeros of the minimal polynomial are cos(5*Pi/9) = - A019819 and cos(7*Pi/9) = - A019859. The mapping z -> 1 - 2*z^2 cyclically permutes the three zeros.  The inverse permutation is given by the mapping z -> 2*z^2 - z - 1. (End)

			0.93969262...

J. E. Maxfield and M. W. Maxfield, Abstract Algebra and Solution by Radicals, Dover Publications ISBN 0-486-67121-6, (1992), p. 197.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Michael Penn, Proof that cos(20°) is irrational, YouTube video, 2022.
Index entries for algebraic numbers, degree 3

Cf. A019819, A019844, A019859, A214778, A332437.

RealDigits[Sin[70 Degree],10,120][[1]] (* Harvey P. Dale, Aug 17 2012 *)

cos(Pi/9) \\ Charles R Greathouse IV, Aug 27 2017

Equals 2*A019844*A019864. - R. J. Mathar, Jan 17 2021
Equals cos(Pi/9) = (1/2)*A332437. - Peter Bala, Oct 21 2021
Equals 2F1(-1/6,1/6 ; 1/2; 3/4). - R. J. Mathar, Aug 31 2025

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

A019884 Decimal expansion of sine of 75 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

Length of one side of the new Type 15 Convex Pentagon. - Michel Marcus, Aug 04 2015

			0.96592582628906828674974319972889736763390483900840455040234307631042...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Casey Mann et al, Type 15 Convex Pentagon, Jul 29 2015.
Wikipedia, Trigonometric constants expressed in real radicals.
Index entries for algebraic numbers, degree 4

Cf. A120683.

RealDigits[Sin[75 Degree], 10, 100][[1]] (* Vincenzo Librandi, Aug 11 2014 *)

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

Equals cos(Pi/12) = [1+sqrt(3)]/[2*sqrt(2)] = A090388 * A020765. - R. J. Mathar, Jun 18 2006
Equals A019859 * A019874 + A019834 * A019849 = A019881 * A019896 + A019812 * A019827 . - R. J. Mathar, Jan 27 2021
Equals 1/(sqrt(6) - sqrt(2)) = 1/A120683. - Amiram Eldar, Aug 04 2022
Largest of the 4 real-valued roots of 16*x^4 -16*x^2 +1=0. - R. J. Mathar, Aug 29 2025
4*this^3 -3*this = A010503. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/8,1/8 ;1/2;3/4) = 2F1(-1/6,1/6;1/2;1/2). - R. J. Mathar, Aug 31 2025

A019864 Decimal expansion of sine of 55 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 12 and denominator 2. - Charles R Greathouse IV, Nov 05 2017

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

RealDigits[Sin[55 Degree],10,120][[1]] (* Harvey P. Dale, Jan 30 2023 *)

sin(11/36*Pi) \\ Charles R Greathouse IV, Nov 05 2017

Equals cos(7*Pi/36) = 2F1(19/24,5/24;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008

Equals A019853 * A019888 + A019820 * A019855 = A019859 * A019894 + A019814 * A019849. - R. J. Mathar, Jan 27 2021

A019938 Decimal expansion of tangent of 40 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the decimal expansion of cotangent of 50 degrees. - Ivan Panchenko, Aug 01 2014

			0.839099631177280011763127298123181364687434283012346533244102...

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

Cf. A019849 (sine of 40 degrees).

SetDefaultRealField(RealField(100)); R:= RealField(); Tan(2*Pi(R)/9); // G. C. Greubel, Nov 25 2018

RealDigits[Tan[2*Pi/9], 10, 100][[1]] (* G. C. Greubel, Nov 25 2018 *)
RealDigits[Tan[40 Degree],10,120][[1]] (* Harvey P. Dale, Apr 06 2022 *)

default(realprecision, 100); tan(2*Pi/9) \\ G. C. Greubel, Nov 25 2018

polrootsreal(x^6-33*x^4+27*x^2-3)[5] \\ Charles R Greathouse IV, Feb 05 2025

numerical_approx(tan(2*pi/9), digits=100) # G. C. Greubel, Nov 25 2018

One of the 6 real-valued roots of x^6-33*x^4+27*x^2-3=0. - R. J. Mathar, Aug 31 2025

Equals A019849/A019859. - R. J. Mathar, Aug 31 2025

A019948 Decimal expansion of tangent of 50 degrees.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Also the decimal expansion of cotangent of 40 degrees. - Ivan Panchenko, Sep 01 2014

			1.19175359259420995870530807186041933693070040770854385365483...

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

Cf. A019859 (sine of 50 degrees).

SetDefaultRealField(RealField(100)); R:= RealField(); Tan(5*Pi(R)/18); // G. C. Greubel, Nov 23 2018

RealDigits[Tan[5*Pi/18], 10, 100][[1]] (* G. C. Greubel, Nov 23 2018 *)

default(realprecision, 100); tan(5*Pi/18) \\ G. C. Greubel, Nov 23 2018

numerical_approx(tan(5*pi/18), digits=100) # G. C. Greubel, Nov 23 2018

A root of 3*x^6 -27*x^4 +33*x^2 -1 =0 (others A019968, A019908). - R. J. Mathar, Aug 29 2025

cp's OEIS Frontend

A019819 Decimal expansion of sine of 10 degrees.

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A019879 Decimal expansion of sine of 70 degrees.

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

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A019884 Decimal expansion of sine of 75 degrees.

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A019864 Decimal expansion of sine of 55 degrees.

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A019938 Decimal expansion of tangent of 40 degrees.

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A019948 Decimal expansion of tangent of 50 degrees.

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