A019845 -id:A019845 - cp's OEIS frontend

A019836 Decimal expansion of sine of 27 degrees.

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

A019866 Decimal expansion of sine of 57 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Nov 05 2017

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 16

RealDigits[Sin[57 Degree],10,120][[1]] (* Harvey P. Dale, Oct 30 2011 *)

sin(19/60*Pi) \\ Charles R Greathouse IV, Nov 05 2017

Equals A019847 * A019880 + A019828 * A019861 = A019863 * A019896 + A019812 * A019845. - R. J. Mathar, Jan 27 2021

A365165 Length of the perimeter of the regular 9-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 24 2023

			6.1563625798620371947937930642806724537...

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A019845 (5-gon), A010487 (4-gon), A365163 (7-gon), A365164 (8-gon), A272488 (edge length).

First[RealDigits[18*Sin[Pi/9], 10, 100]] (* Paolo Xausa, Mar 19 2024 *)

Equals 9*A272488.

A019848 Decimal expansion of sine of 39 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This sequence is also decimal expansion of cosine of 51 degrees. - Mohammad K. Azarian, Jun 29 2013

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Nov 05 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

Cf. A019812, A019822, A019835, A019845, A019886, A019873, A019896, A019863.

RealDigits[Sin[39 Degree],10,120][[1]] (* Harvey P. Dale, Oct 02 2015 *)

sin(13/60*Pi) \\ Charles R Greathouse IV, Nov 05 2017

Equals A019835*A019886 + A019822*A019873 = A019845*A019896 + A019812*A019863. - R. J. Mathar, Jan 27 2021

A158934 Decimal expansion of xi = (cos(Pi/5) - 1/2) / (sin(Pi/5) + 1/2).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Mar 31 2009

This constant xi arises in the Davenport-Heilbronn zeta-function Z(s)=Sum_{k>=1} b(k)/k^s where b(k) is the 5-periodic sequence with period [1,xi,-xi,0]. Z satisfies a functional equation (like zeta) but does not satisfy RH. Some nontrivial zeros are off the critical line (see reference).

			0.2840790438404122960282...

Peter Borwein, Stephen Choi, Brendan Rooney and Andrea Weirathmueller, The Riemann Hypothesis, Springer, 2009, pp. 136-137.

Bruce C. Berndt, Heng Huat Chan and Liang-Cheng Zhang, Explicit evaluations of the Rogers-Ramanujan continued fraction, Journal für die reine und angewandte Mathematik, Vol. 480 (1996), pp. 141-160, eq. (1.1).
Harold Davenport and Hans Heilbronn, On the zeros of certain Dirichlet series, Journal of the London Mathematical Society, Vol. s1-11, No. 3 (1936), pp. 181-185.
Harold Davenport and Hans Heilbronn, On the zeros of certain Dirichlet series (Second paper), Journal of the London Mathematical Society, Vol. s1-11, No. 4 (1936), pp. 307-312.

Cf. A001622, A158241, A188593, A019845.

(Sqrt[5]-1) / (2+Sqrt[10-2*Sqrt[5]]) // RealDigits[#, 10, 104]& // First (* Jean-François Alcover, Mar 04 2013 *)

PARI
```
xi=(cos(Pi/5)-1/2)/(sin(Pi/5)+1/2)
```

Equals (sqrt(10-2*sqrt(5))-2)/(sqrt(5)-1).
Equals (A001622-1)/(2*A019845+1). - R. J. Mathar, Apr 02 2009
Equals sqrt((5 + sqrt(5))/2) - (sqrt(5) + 1)/2 = A188593 - A001622. - Amiram Eldar, Jan 23 2022

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
Equals A232738/(2*A343056). - R. J. Mathar, Sep 05 2025

A365163 Length of the perimeter of the regular heptagon with unit circumradius.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 24 2023

			6.0743723476458136866607566598...

Cf. A019845 (5-gon), A010487 (4-gon), A365164 (8-gon), A365165 (9-gon), A272487 (edge length), A104957 (area).

f[R_, s_] := 2*R*s*Sin[Pi/s]; a[n_] := RealDigits[f[1, 7], 10, n][[1]]; a[92] (* Robert P. P. McKone, Aug 24 2023 *)

Equals 7*A272487.

A365164 Length of the perimeter of the regular octagon with unit circumradius.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Aug 24 2023

			6.122934917841436347655359744486...

Cf. A019845 (5-gon), A010487 (4-gon), A365163 (7-gon), A365165 (9-gon), A101464 (edge length), A010466 (area).

f[R_, s_] := 2*R*s*Sin[Pi/s]; a[n_] := RealDigits[f[1, 8], 10, n][[1]]; a[109] (* Robert P. P. McKone, Aug 24 2023 *)

Equals 8*A101464.

A371604 Decimal expansion of 5 * sqrt(3 - phi) / (2 * Pi).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 01 2024

			0.93548928378863903321291906615298281...

Cf. A019845, A060294, A086089, A089491, A094874, A112628, A182007, A258403.

RealDigits[5 Sqrt[3 - GoldenRatio]/(2 Pi), 10, 99][[1]]

Equals Product_{k>=1} (1 - 1/(5*k)^2).

Equals A258403/Pi. - Hugo Pfoertner, Apr 01 2024

A348757 Decimal expansion of the area of a regular pentagram inscribed in a unit-radius circle.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Nov 12 2021

An algebraic number of degree 4. The smaller of the two positive roots of the equation 16*x^4 - 2500*x^2 + 3125 = 0.

			1.12256994144896343110486287949381696894803120580270...

Robert B. Banks, Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics, Princeton University Press, 2012, p. 15.

Herta T. Freitag, Problem 3855, School Science and Mathematics, Vol. 81, No. 4 (1981), p. 352; Solution to Problem 3855 by David A. Blaeuer, ibid., Vol. 82, No. 3 (1982), pp. 265-266.
Mathematics Stack Exchange, Area of a five pointed star, 2014.
Wikipedia, Pentagram.
Index entries for algebraic numbers, degree 4

Cf. A001622, A019845, A019916, A019952, A019970, A094874, A104955, A134974, A179050, A188593, A344172.

RealDigits[5*Sin[Pi/5]/GoldenRatio^2, 10, 100][[1]]

Equals 5*sin(Pi/5)/phi^2, where phi is the golden ratio (A001622).
Equals 5/(cot(Pi/5) + cot(Pi/10)).
Equals 10*tan(Pi/10)/(3 - tan(Pi/10)^2).
Equals (5/2)*sqrt((25 -11*sqrt(5))/2).
Equals 5*(5 - sqrt(5))/(4*sqrt(5 + 2*sqrt(5))) = A094874 * A179050 = 10 * A094874 / A344172.

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A019836 Decimal expansion of sine of 27 degrees.

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A019866 Decimal expansion of sine of 57 degrees.

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A365165 Length of the perimeter of the regular 9-gon with unit circumradius.

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A019848 Decimal expansion of sine of 39 degrees.

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A158934 Decimal expansion of xi = (cos(Pi/5) - 1/2) / (sin(Pi/5) + 1/2).

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

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A365163 Length of the perimeter of the regular heptagon with unit circumradius.

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A365164 Length of the perimeter of the regular octagon with unit circumradius.