A019814 -id:A019814 - 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

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

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

A019874 Decimal expansion of sine of 65 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 12 and denominator 2. - Charles R Greathouse IV, Nov 06 2017

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

RealDigits[Sin[65 Degree],10,120][[1]] (* Harvey P. Dale, Oct 29 2020 *)

sin(13/36*Pi) \\ Charles R Greathouse IV, Nov 06 2017

Equals cos(5*Pi/36) = 2F1(17/24,7/24;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008

Equals A019861 * A019886 + A019822 * A019847 = A010527 * A019894 + A019814*(1/2). - 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

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

A019844 Decimal expansion of sine of 35 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This sequence is also decimal expansion of cosine of 55 degrees. - Mohammad K. Azarian, Jun 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

RealDigits[Sin[35 Degree],10,120][[1]] (* Harvey P. Dale, Mar 26 2013 *)

sin(35*Pi/180) \\ Michel Marcus, Aug 11 2014

Equals A019837 * A019892 + A019816 * A019871 = (1/2)* A019894 + A019814 * A010527. - R. J. Mathar, Jan 27 2021

A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Jan 09 2022

			0.09801714032956060199419...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Penn, The exact value of sin 5⅝°??, YouTube video, 2021.
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 16

sin(Pi/m): A010527 (m=3), A010503 (m=4), A019845 (m=5), A323601 (m=7), A182168 (m=8), A019829 (m=9), A019827 (m=10), A019824 (m=12), A232736 (m=14), A019821 (m=15), A232738 (m=16), A241243 (m=17), A019819 (m=18), A019818 (m=20), A343054 (m=24), A019815 (m=30), this sequence (m=32), A019814 (m=36).

RealDigits[Sin[Pi/32], 10, 100, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
imag(I^(1/16))
```
PARI
```
sin(Pi/32)
```
PARI
```
sqrt(2-sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(sin(pi/32), digits=123) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2+sqrt(2)))).

One of the 16 real roots of -128*x^2 +2688*x^4 -21504*x^6 +84480*x^8 +32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +1 =0. - R. J. Mathar, Aug 29 2025

cp's OEIS Frontend

A019819 Decimal expansion of sine of 10 degrees.

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A019824 Decimal expansion of sine of 15 degrees.

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A019894 Decimal expansion of sine of 85 degrees.

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A019874 Decimal expansion of sine of 65 degrees.

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A019834 Decimal expansion of sine of 25 degrees.

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A019864 Decimal expansion of sine of 55 degrees.

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A019844 Decimal expansion of sine of 35 degrees.

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A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

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