A019875 -id:A019875 - cp's OEIS frontend

A019815 Decimal expansion of sine of 6 degrees.

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

A019857 Decimal expansion of sine of 48 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

			0.74314482...

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

Cf. A019833, A019875.

RealDigits[ Sin[4 Pi/15], 10, 111] (* Robert G. Wilson v *)

sin(4*Pi/15) \\ Charles R Greathouse IV, Aug 27 2017

Equals cos(7*pi/30) = 2F1(17/20,3/20;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
Equals 2*A019833*A019875. - R. J. Mathar, Jan 17 2021
Equals 1/(sqrt(5+2*sqrt(5))-sqrt(3)). - Seiichi Manyama, Mar 19 2021
4*this^3 -3*this = -A019845. - R. J. Mathar, Aug 29 2025
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

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

A019836 Decimal expansion of sine of 27 degrees.

Original entry on oeis.org

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

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, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 8

RealDigits[Sin[27 Degree],10,120][[1]] (* Harvey P. Dale, Nov 09 2014 *)

cos(7*Pi/20) \\ Charles R Greathouse IV, Aug 27 2017

Equals A019827 * A019890 + A019818 * A019881 = A019833 * A019896 + A019812 * A019875. - R. J. Mathar, Jan 27 2021

Equals cos(7*Pi/20). 2*this^2-1 = -A019845. - R. J. Mathar, Aug 29 2025

A019878 Decimal expansion of sine of 69 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Nov 06 2017

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 16

RealDigits[Sin[69 Degree],10,120][[1]] (* Harvey P. Dale, Dec 19 2011 *)

sin(23/60*Pi) \\ Charles R Greathouse IV, Nov 06 2017

Equals A019855 * A019876 + A019832 * A019853 = A019875 * A019896 + A019812 * A019833. - R. J. Mathar, Jan 27 2021

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

A306603 a(n) = (2 cos(Pi/15))^n + (2 cos(7 Pi/15))^n + (2 cos(11 Pi/15))^n + (2 cos(13 Pi/15))^n.

Original entry on oeis.org

4, -1, 9, -1, 29, 4, 99, 34, 349, 179, 1254, 824, 4559, 3574, 16704, 15004, 61549, 61709, 227799, 250229, 846254, 1004149, 3153984, 3997399, 11788879, 15812504, 44178624, 62229509, 165946124, 243873904, 624650004, 952400599, 2355748909, 3708579599

Offset: 0

Greg Dresden, Feb 27 2019

a(n) is obtained from the Girard-Waring formula for the sum of powers of N = 4 indeterminates (see A324602), with the elementary symmetric functions e_1 = -1, e_2 = -4, e_3 = -4 and e_4 = 1. The arguments are e_j(x_1, x_2, x_3, x_4), for j = 1..4, with the zeros {x_i}A187360,%20for%20n%20=%2015),%20appearing%20to%20the%20power%20n%20in%20the%20formula%20given%20above.%20-%20_Wolfdieter%20Lang">{i=1..4} of the minimal polynomial of 2*cos(Pi/15) (see A187360, for n = 15), appearing to the power n in the formula given above. - _Wolfdieter Lang, May 08 2019

Index entries for linear recurrences with constant coefficients, signature (-1,4,4,-1).

Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A187360, A324602.

Table[Sum[(2.0 Cos[k Pi/15])^n, {k, {1, 7, 11, 13}}] // Round, {n, 1, 30}]
LinearRecurrence[{-1,4,4,-1},{4,-1,9,-1},40] (* Harvey P. Dale, Jun 02 2024 *)

G.f.: (4*x^3+8*x^2-3*x-4)/(-x^4+4*x^3+4*x^2-x-1). - Alois P. Heinz, Feb 27 2019

a(n) = -a(n-1) + 4*a(n-2) + 4*a(n-3) -a(n-4). - Greg Dresden, Feb 27 2019

A019964 Decimal expansion of tangent of 66 degrees.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Also the decimal expansion of cotangent of 24 degrees. - Ivan Panchenko, Sep 01 2014

			2.24603677390421605416332143841640915914036310102689708141...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Wikipedia, Exact trigonometric constants

Cf. A019875 (sine 66 degrees).

SetDefaultRealField(RealField(100)); R:= RealField(); Tan(11*Pi(R)/30); // G. C. Greubel, Nov 21 2018

RealDigits[Tan[73*Pi/180], 10, 100][[1]] (* G. C. Greubel, Nov 21 2018 *)

default(realprecision, 100); tan(11*Pi/30) \\ G. C. Greubel, Nov 21 2018

numerical_approx(tan(11*pi/30), digits=100) # G. C. Greubel, Nov 21 2018

A306610 a(n) = (2cos(Pi/15))^(-n) + (2cos(7Pi/15))^(-n) + (2cos(11Pi/15))^(-n) + (2cos(13*Pi/15))^(-n), for n >= 1.

Original entry on oeis.org

4, 24, 109, 524, 2504, 11979, 57299, 274084, 1311049, 6271254, 29997829, 143491199, 686373809, 3283190949, 15704770004, 75121978804, 359337430474, 1718849676159, 8221921677724, 39328626006254, 188124003629279, 899869747188249, 4304424455586134

Offset: 1

Greg Dresden, Feb 28 2019

-a(n) is the coefficient of x in the minimal polynomial for (2*cos(Pi/15))^n, for n >= 1. The coefficients of -x^3 are A306603(n), and those of x^2 are A306611(n).

a(n) is obtained from the Girard-Waring formula for the sum of powers of N = 4 indeterminates (see A324602), with the elementary symmetric functions e_1 = 4, e_2 = -4, e_3 = -1 and e_4 = 1. The arguments are e_j(1/x_1, 1/x_2, 1/x_3, 1/x_4), for j = 1..4, with the zeros {x_i}{i=1..4} of the minimal polynomial of 2*cos(Pi/15), appearing under the negative powers of the formula given above. - _Wolfdieter Lang, May 08 2019

Index entries for linear recurrences with constant coefficients, signature (4,4,-1,-1).

Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A306603 (positive powers of these cosines), A306611, A324602.

Table[Round[N[Sum[(2 Cos[k Pi/15])^(-n), {k,{1,7,11,13}}],50]],{n,1,30}]

a(n) = 4a(n-1) + 4a(n-2) - a(n-3) - a(n-4).
G.f.: x*(-4x^3 -3x^2 +8x +4)/(x^4 +x^3 -4x^2 -4x +1).
a(n) = round((2*cos(7*Pi/15))^(-n)) for n >= 3.

A307886 Array of coefficients of the minimal polynomial of (2*cos(Pi/15))^n, for n >= 1 (ascending powers).

Original entry on oeis.org

1, -4, -4, 1, 1, 1, -24, 26, -9, 1, 1, -109, -49, 1, 1, 1, -524, 246, -29, 1, 1, -2504, -619, -4, 1, 1, -11979, 2621, -99, 1, 1, -57299, -7774, -34, 1, 1, -274084, 30126, -349, 1, 1, -1311049, -97879, -179, 1, 1, -6271254, 363131, -1254, 1, 1, -29997829, -1237504, -824, 1

Offset: 1

Greg Dresden and Wolfdieter Lang, May 02 2019

The length of each row is 5.

The minimal polynomial of (2*cos(Pi/15))^n, for n >= 1, is C(15, n, x) = Product_{j=0..3} (x - (x_j)^n) = Sum_{k=0} T(n, k) x^k, where x_0 = 2*cos(Pi/15), x_1 = 2*cos(7*Pi/15), x_2 = 2*cos(11*Pi/15), and x_3 = 2*cos(13*Pi/15) are the zeros of C(15, 1, x) with coefficients given in A187360 (row n=15).

			The rectangular array T(n, k) begins:
n\k 0      1      2      3      4
---------------------------------
1:  1     -4     -4      1      1
2:  1    -24     26     -9      1
3:  1   -109    -49      1      1
4:  1   -524    246    -29      1
5:  1  -2504   -619     -4      1
6:  1 -11979   2621    -99      1
7:  1 -57299  -7774    -34      1
...

Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A187360, A306603, A306610, A306611.

Flatten[Table[CoefficientList[MinimalPolynomial[(2*Cos[\[Pi]/15])^n, x], x], {n, 1, 15}]]

T(n,k) = the coefficient of x^k in C(15, n, x), n >= 1, k=0,1,2,3,4, with C(15, n, k) the minimal polynomial of (2*cos(Pi/15))^n, for n >= 1 as defined above.

T(n, 0) = T(n, 4) = 1. T(n, 1) = -A306610(n), T(n, 2) = A306611(n), T(n, 3) = -A306603(n), n >= 1.

cp's OEIS Frontend

A019815 Decimal expansion of sine of 6 degrees.

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A019857 Decimal expansion of sine of 48 degrees.

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A019833 Decimal expansion of sine of 24 degrees.

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A019836 Decimal expansion of sine of 27 degrees.

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A019878 Decimal expansion of sine of 69 degrees.

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A019886 Decimal expansion of sine of 77 degrees.

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A306603 a(n) = (2 cos(Pi/15))^n + (2 cos(7 Pi/15))^n + (2 cos(11 Pi/15))^n + (2 cos(13 Pi/15))^n.

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A019964 Decimal expansion of tangent of 66 degrees.

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A306610 a(n) = (2cos(Pi/15))^(-n) + (2cos(7Pi/15))^(-n) + (2cos(11Pi/15))^(-n) + (2cos(13*Pi/15))^(-n), for n >= 1.