A019898 -id:A019898 - cp's OEIS frontend

A019812 Decimal expansion of sine of 3 degrees.

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Nov 02 2012
The minimal polynomial is p(x) = 65536*x^16 - 262144*x^14 + 430080*x^12 - 372736*x^10 + 182784*x^8 - 50176*x^6 + 7040*x^4 - 384*x^2 + 1 and p(x/2) = x^16 - 16*x^14 + 105*x^12 - 364*x^10 + 714*x^8 - 784*x^6 + 440*x^4 - 96*x^2 + 1. - Joerg Arndt, Apr 23 2024
The Fifteenth Century Persian mathematician Jamshid Al-Kashi was the first to calculate the value of sine of one degree correct to ten sexagesimal places (17 decimal digits) from sine of 3 degrees in his Risala al-Watar wa'l Jaib. - Mohammad K. Azarian,Jan 14 2017

			0.052335956242943832722118629609...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Mohammad K. Azarian, A Study of Risa-la al-Watar wa'l Jaib ("The Treatise on the Chord and Sine"), Forum Geometricorum, Volume 15 (2015) 229-242. Mathematical Reviews, MR 3418854 (Reviewed), Zentralblatt MATH, Zbl 1328.01015.
Dr. Rob, The Math Forum at Drexel, The Exact Value of the Sine of 1 Degree
Scott Surgent, Exact values of the sine and cosine functions in increments of 3 degrees, 2012
Wikipedia, Exact trigonometric values
Index entries for algebraic numbers, degree 16

Cf. A019896 (cosine), A019901 (tangent), A019985 (cotangent): for 3 degrees.

Cf. A019810, A019811, A019897, A019898.

First[RealDigits[Sin[Pi/60], 10, 100, -1]] (* Paolo Xausa, Jun 19 2024 *)

sin(Pi/60) \\ Charles R Greathouse IV, Aug 27 2017

Equals cos(87 degrees) = cos(29*Pi/60) = sin(Pi/60) = sqrt(8-sqrt(3)-sqrt(15)-sqrt(10-2*sqrt(5)))/4 (an intermediate calculation by Dr. Rob - see Math Forum link). - Rick L. Shepherd, Jul 03 2006

Equals A019811*A019898 + A019810*A019897. - R. J. Mathar, Jan 27 2021

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

A019816 Decimal expansion of sine of 7 degrees.

Original entry on oeis.org

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

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[7 Degree],10,120][[1]] (* Harvey P. Dale, Mar 25 2016 *)

sin(7*Pi/180) \\ Charles R Greathouse IV, Aug 27 2017

Also cosine of 83 degrees. - Charles R Greathouse IV, Aug 27 2017

Equals A019815 * A019898 + A019810 * A019893. - R. J. Mathar, Jan 27 2021

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

A019811 Decimal expansion of sine of 2 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

			0.034899...

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

sin(Pi/90) \\ Charles R Greathouse IV, Aug 27 2017

Equals 2*A019810*A019898. - R. J. Mathar, Jan 17 2021

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

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

A019826 Decimal expansion of sine of 17 degrees.

Original entry on oeis.org

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

sin(17*Pi/180) \\ Charles R Greathouse IV, Aug 27 2017

Equals A019825 * A019898 + A019810 * A019883. - R. J. Mathar, Jan 27 2021

A111493 Decimal expansion of the square of tan 1 degree.

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Nov 21 2005

An algebraic integer with degree 24. - Charles R Greathouse IV, Aug 27 2017

			0.000304679291648291592499514192263702726536635855843973780688562133....

Index entries for algebraic numbers, degree 24

Cf. A019810, A019898.

tan(Pi/180)^2 \\ Charles R Greathouse IV, Aug 27 2017

Equals A111486/A111590. - R. J. Mathar, Aug 29 2025

A111587 a(n) = 2a(n-1) + 2a(n-3) + a(n-4), a(0) = 1, a(1) = 4, a(2) = 9, a(3) = 20.

Original entry on oeis.org

1, 4, 9, 20, 49, 120, 289, 696, 1681, 4060, 9801, 23660, 57121, 137904, 332929, 803760, 1940449, 4684660, 11309769, 27304196, 65918161, 159140520, 384199201, 927538920, 2239277041, 5406093004, 13051463049, 31509019100, 76069501249

Offset: 0

Creighton Dement, Aug 08 2005

Let (b(n)) be the p-INVERT of (1,2,2,2,2,2,...) using p(S) = 1 - S^2; then
b(0) = 0 and b(n) = a(n-1) for n >= 1; see A292400. - Clark Kimberling, Sep 30 2017
Floretion Algebra Multiplication Program, FAMP Code: 2kbasekseq[J+G] with J = + j' + k' + 'ii' and G = + .5'ii' + .5'jj' + .5'kk' + .5e

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,2,1).

Cf. A019898, A046729, A090390, A111588.

I:=[1,4,9,20]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-3)+Self(n-4): n in [1..35]]; // Vincenzo Librandi, Oct 01 2017

LinearRecurrence[{2,0,2,1},{1,4,9,20},30] (* Harvey P. Dale, Jul 26 2011 *)
CoefficientList[Series[(x + 1)^2 / ((x^2 + 1) (1 - 2 x - x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Oct 01 2017 *)

a(2n) = A090390(n+1), a(2n+1) = A046729(n+1);
G.f.: (x+1)^2/((x^2+1)*(1-2*x-x^2)). [sign flipped by R. J. Mathar, Nov 10 2009]
a(n) = A057077(n+1)/2 - A001333(n+2)/2. - R. J. Mathar, Nov 10 2009

A111761 Decimal expansion of (tan 1 degree)^(1/3).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Nov 21 2005

Algebraic integer of degree 72. - Charles R Greathouse IV, May 13 2019

			0.2594023002497509797486867779110501206090479682965121888659767933996
55599129823201180630419298561199613626894277991846295

Cf. A019810, A019898.

sqrtn(tan(Pi/180),3) \\ Charles R Greathouse IV, May 13 2019

Offset corrected by Mohammad K. Azarian, Dec 11 2008

cp's OEIS Frontend

A019812 Decimal expansion of sine of 3 degrees.

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A019814 Decimal expansion of sine of 5 degrees.

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A019816 Decimal expansion of sine of 7 degrees.

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A019820 Decimal expansion of sine of 11 degrees.

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A019811 Decimal expansion of sine of 2 degrees.

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A019822 Decimal expansion of sine of 13 degrees.

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A019826 Decimal expansion of sine of 17 degrees.

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A111493 Decimal expansion of the square of tan 1 degree.

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A111587 a(n) = 2a(n-1) + 2a(n-3) + a(n-4), a(0) = 1, a(1) = 4, a(2) = 9, a(3) = 20.