A019821 -id:A019821 - cp's OEIS frontend

A019887 Decimal expansion of sine of 78 degrees.

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

Offset: 0

N. J. A. Sloane

Equals sin(13*Pi/30). - Wesley Ivan Hurt, Aug 31 2014

A quartic number with denominator 2 and minimal polynomial 16x^4 + 8x^3 - 16x^2 - 8x + 1. - Charles R Greathouse IV, Aug 27 2017

			0.9781476007338056379285667478695995324597378088626771078851...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 4

Digits:=100: evalf(sin(13*Pi/30)); # Wesley Ivan Hurt, Aug 31 2014

First@RealDigits[Sin[78 Degree], 10, 120] (* Michael De Vlieger, Aug 31 2014 *)

cos(Pi/15) \\ Charles R Greathouse IV, Aug 27 2017

Equals cos(Pi/15) = [sqrt(5)-1]*[1+sqrt(3)*sqrt{5+2*sqrt(5)}]/8 = [A002163-1]*[1+A002194*A019970]/8. - R. J. Mathar, Jun 18 2006
Equals 2*A019848*A019860. - R. J. Mathar, Jan 17 2021
4*this^3 -3*this = A019863. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/10,1/10 ; 1/2 ; 3/4). - R. J. Mathar, Aug 31 2025
A root of 16*x^4+8*x^3-16*x^2-8*x+1=0. - R. J. Mathar, Aug 31 2025
Equals A019833/(2*A019821). - R. J. Mathar, Sep 06 2025

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
Equals 1/(4*A019884). - R. J. Mathar, Sep 05 2025

A019875 Decimal expansion of sine of 66 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

A quartic number with denominator 2. - Charles R Greathouse IV, Aug 27 2017

			0.913545457...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 4

Cf. A019821, A019842, A019866, A019887.

RealDigits[N[Cos[2*Pi/15], 110]][[1]] (* Wesley Ivan Hurt, Dec 27 2021 *)

cos(2*Pi/15) \\ Charles R Greathouse IV, Aug 27 2017

Equals cos(2*Pi/15) = 2*A019887^2 - 1 = 1 - 2*A019821^2. - R. J. Mathar, Jun 18 2006
Equals 2*A019842*A019866. - R. J. Mathar, Jan 17 2021
Largest of the 4 real-valued roots of 16*x^4-8*x^3-16*x^2+8*x+1=0. (Other A019851, -A019815, -A019887). - R. J. Mathar, Sep 04 2025

A019815 Decimal expansion of sine of 6 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Decimal expansion of 1/8 (-1 - sqrt(5) + sqrt(6*(5 - sqrt(5)))). - Artur Jasinski, Oct 28 2008

			sin(Pi/30) = 0.10452846...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 4

First[RealDigits[1/8 (-1 - Sqrt[5] + Sqrt[6 (5 - Sqrt[5])]), 10, 100]] (* Artur Jasinski, Oct 28 2008 *)
RealDigits[Sin[6 Degree],10,120][[1]] (* Harvey P. Dale, Apr 27 2012 *)

sin(Pi/30) \\ Charles R Greathouse IV, Jan 30 2016

Equals cos(7*Pi/15) = -cos(8*Pi/15) = 2F1(6/5,-1/5;1/2;3/4) / 2 = -2F1(13/10,-3/10;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
Equals 2*A019812*A019896. - R. J. Mathar, Jan 17 2021
Smallest positive of the 4 real-valued roots of 16*x^4+8*x^3-16*x^2-8*x+1=0. (Other A019887, -A019875, -A019851) - R. J. Mathar, Aug 31 2025
Equals A019821/(2*A019893). - R. J. Mathar, Sep 05 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

A272534 Decimal expansion of the edge length of a regular 15-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 02 2016

15-gon is the first m-gon with odd composite m which is constructible (see A003401) in virtue of the fact that 15 is the product of two distinct Fermat primes (A019434). The next such case is 51-gon (m=3*17), followed by 85-gon (m=5*17), 771-gon (m=3*257), etc.
From Wolfdieter Lang, Apr 29 2018: (Start)
This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Viète. See the Havil reference, problem 4, pp. 69-74. See also the comments in A302711 with a link to Romanus' book, Exemplum quaesitum.
This problem is equivalent to R(45, 2*sin(Pi/675)) = 2*sin(Pi/15), with a special case of monic Chebyshev polynomials of the first kind, named R, given in A127672. For the constant 2*sin(Pi/675) see A302716. (End)

			0.415823381635518674203484568810250332433169521255447672814363947...

Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for sequences related to Chebyshev polynomials.
Index entries for algebraic numbers, degree 8.

Cf. A003401, A019434, A127672, A302711, A302716.

Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272535 (16), A228787 (17), A272536 (20).

RealDigits[N[2Sin[Pi/15], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/15)
```

Equals 2*sin(Pi/m) for m=15, 2*A019821.
Also equals (sqrt(3) - sqrt(15) + sqrt(10 + 2*sqrt(5)))/4.
Also equals sqrt(7 - sqrt(5) - sqrt(30 - 6*sqrt(5)))/2. This is the rewritten expression of the Havil reference on top of p. 70. - Wolfdieter Lang, Apr 29 2018

A019833 Decimal expansion of sine of 24 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

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

sin(2*Pi/15) \\ Charles R Greathouse IV, Aug 27 2017

Equals sin(2*Pi/15) = sqrt(1-A019875^2) = 2*A019821*A019887. - R. J. Mathar, Jun 18 2006

One of the 8 real-valued roots of 256*x^8-448*x^6+224*x^4-32*x^2+1=0. - R. J. Mathar, Aug 31 2025

A019822 Decimal expansion of sine of 13 degrees.

Original entry on oeis.org

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

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

RealDigits[Sin[13 Degree],10,120][[1]] (* Harvey P. Dale, Sep 29 2013 *)

sin(13*Pi/180) \\ Charles R Greathouse IV, Aug 27 2017

Equals A019821 * A019898 + A019810 * A019887. - R. J. Mathar, Jan 27 2021

A019904 Decimal expansion of tangent of 6 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the decimal expansion of cotangent of 84 degrees. - Mohammad K. Azarian, Jun 30 2013
An algebraic integer of degree 8. - Charles R Greathouse IV, Nov 17 2017
Root of the polynomial 1-92*x^2+134*x^4-28*x^6+x^8=0. (Others: A019940, A019964, A019976). - R. J. Mathar, Sep 06 2025

			tan(Pi/30) = 0.105104235...

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

RealDigits[Tan[6 Degree],10,120][[1]] (* Harvey P. Dale, Jun 18 2024 *)

tan(Pi/30) \\ Charles R Greathouse IV, Nov 17 2017

Equals A019815 / A019839 = A019821/(1+A019887). - R. J. Mathar, Sep 06 2025

A019976 Decimal expansion of tangent of 78 degrees.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Also the decimal expansion of cotangent of 12 degrees. - Ivan Panchenko, Sep 01 2014

An algebraic integer of degree 8. - Charles R Greathouse IV, Nov 05 2017

			tan(13*Pi/30) = 4.7046301...

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

1/tan(Pi/15) \\ Charles R Greathouse IV, Nov 05 2017

Equals A019887 / A019821 = A019833/(1-A019875) . - R. J. Mathar, Sep 06 2025

cp's OEIS Frontend

A019887 Decimal expansion of sine of 78 degrees.

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A019824 Decimal expansion of sine of 15 degrees.

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A019875 Decimal expansion of sine of 66 degrees.

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A019815 Decimal expansion of sine of 6 degrees.

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A019890 Decimal expansion of sine of 81 degrees.

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A272534 Decimal expansion of the edge length of a regular 15-gon with unit circumradius.

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A019833 Decimal expansion of sine of 24 degrees.

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