A019890 -id:A019890 - cp's OEIS frontend

A019827 Decimal expansion of sin(Pi/10) (angle of 18 degrees).

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

Offset: 0

N. J. A. Sloane

Decimal expansion of cos(2*Pi/5) (angle of 72 degrees).
Also the imaginary part of i^(1/5). - Stanislav Sykora, Apr 25 2012
One of the two roots of 4x^2 + 2x - 1 (the other is the sine of 54 degrees times -1 = -A019863). - Alonso del Arte, Apr 25 2015
This is the height h of the isosceles triangle in a regular pentagon inscribed in a unit circle, formed by a diagonal as base and two adjacent radii. h = cos(2*Pi/5) = sin(Pi/10). - Wolfdieter Lang, Jan 08 2018
Quadratic number of denominator 2 and minimal polynomial 4x^2 + 2x - 1. - Charles R Greathouse IV, May 13 2019
Largest superstable width of the logistic map (see Finch). - Stefano Spezia, Nov 23 2024

			0.30901699437494742410229341718281905886015458990288143106772431135263...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.9 and 8.19, pp. 66, 535.

Zak Seidov, Table of n, a(n) for n = 0..999
Hideyuki Ohtsuka, Problem B-1237, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 56, No. 4 (2018), p. 366; A Telescoping Product, Solution to Problem B-1237 by Steve Edwards, ibid., Vol. 57, No. 4 (2019), pp. 369-370.
Wikipedia, Exact trigonometric constants.
Index entries for algebraic numbers, degree 2.

Cf. A001622, A019845, A019863, A055588, A094214, A134945, A187798, A341332, A377697.

RealDigits[Sin[18 Degree], 10, 108][[1]] (* Alonso del Arte, Apr 20 2015 *)

sin(Pi/10) \\ Charles R Greathouse IV, Feb 03 2015

polrootsreal(4*x^2 + 2*x - 1)[2] \\ Charles R Greathouse IV, Feb 03 2015

Equals (sqrt(5) - 1)/4 = (phi - 1)/2 = 1/(2*phi), with phi from A001622.
Equals 1/(1 + sqrt(5)). - Omar E. Pol, Nov 15 2007
Equals 1/A134945. - R. J. Mathar, Jan 17 2021
Equals 2*A019818*A019890. - R. J. Mathar, Jan 17 2021
Equals Product_{k>=1} 1 - 1/(phi + phi^k), where phi is the golden ratio (A001622) (Ohtsuka, 2018). - Amiram Eldar, Dec 02 2021
Equals Product_{k>=1} (1 - 1/A055588(k)). - Amiram Eldar, Nov 28 2024
Equals A094214/2 = 1-A187798 = A341332/Pi = (A377697-2)/3. - Hugo Pfoertner, Nov 28 2024
This^2 + A019881^2 = 1. - R. J. Mathar, Aug 31 2025

A019818 Decimal expansion of sine of 9 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the imaginary part of i^(1/10). - Stanislav Sykora, Apr 25 2012

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

			sin(Pi/20) = 0.1564344650402308690101053194671668923138...

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

Cf. A019812, A019815, A019893, A019896.

RealDigits[Sin[9 Degree],10,120][[1]] (* Harvey P. Dale, Sep 15 2014 *)

sin(Pi/20) \\ Charles R Greathouse IV, Aug 27 2017

sqrt(8-2*sqrt(10+2*sqrt(5)))/4 = sqrt((1/2)*(1 - sqrt((1/8)*(5 + sqrt(5))))). - Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 13 2006
Equals A019815*A019896 + A019812*A019893. - R. J. Mathar, Jan 27 2021
This^2 + A019890^2=1. - R. J. Mathar, Aug 31 2025
Smallest positive of the 8 real-valued roots of 256*x^8 -512*x^6 +304*x^4 -48*x^2+1=0. - R. J. Mathar, Aug 31 2025

A232735 Decimal expansion of the real part of I^(1/7), or cos(Pi/14).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding imaginary part is in A232736.

Root of the equation -7 + 56*x^2 - 112*x^4 + 64*x^6 = 0. - Vaclav Kotesovec, Apr 04 2021

			0.974927912181823607018131682993931217232785800619997437648...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
A. Arman, A. Bondarenko, and A. Prymak, Convex bodies of constant width with exponential illumination number, arXiv:2304.10418 [math.MG], 2023.
Index entries for algebraic numbers, degree 6

Cf. A232736 (imaginary part), A010503 (real(I^(1/2))), A010527 (real(I^(1/3))), A144981 (real(I^(1/4))), A019881 (real(I^(1/5))), A019884 (real(I^(1/6))), A232737 (real(I^(1/8))), A019889 (real(I^(1/9))), A019890 (real(I^(1/10))).

R:= RealField(100); Cos(Pi(R)/14); // G. C. Greubel, Sep 19 2022

RealDigits[Cos[Pi/14],10,120][[1]] (* Harvey P. Dale, Dec 15 2018 *)

numerical_approx(cos(pi/14), digits=120) # G. C. Greubel, Sep 19 2022

2*this^2 -1 = A073052. - R. J. Mathar, Aug 29 2025

Equals 2F1(-1/14,1/14;1/2;1) . - 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

A019872 Decimal expansion of sine of 63 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Nov 06 2017

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

RealDigits[Sin[63 Degree],10,120][[1]] (* Harvey P. Dale, Oct 31 2018 *)

sin(31/60*Pi) \\ Charles R Greathouse IV, Nov 06 2017

Equals A019851 * A019878 + A019830 * A019857 = A010527 * A019896 + A019812 * (1/2). - R. J. Mathar, Jan 27 2021
This^2 + A019836^2=1. - R. J. Mathar, Aug 31 2025
One of the 8 real-valued roots of 256*x^8-512*x^6+304*x^4-48*x^2+1=0. (Other A019890, A019836, A019818) - R. J. Mathar, Aug 31 2025

A232737 Decimal expansion of the real part of I^(1/8), or cos(Pi/16).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding imaginary part is in A232738.

			0.9807852804032304491261822361342390369739337308933360950029160885453...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Wikipedia, Trigonometric constants expressed in real radicals.
Index entries for algebraic numbers, degree 8

Cf. A232738 (imaginary part), A010503 (real(I^(1/2))), A010527 (real(I^(1/3))), A144981 (real(I^(1/4))), A019881 (real(I^(1/5))), A019884 (real(I^(1/6))), A232735 (real(I^(1/7))), A019889 (real(I^(1/9))), A019890 (real(I^(1/10))).

RealDigits[Cos[Pi/16], 10, 120][[1]] (* Amiram Eldar, Jun 29 2023 *)

real(I^(1/8)) \\ Seiichi Manyama, Apr 04 2021

cos(Pi/16) \\ Seiichi Manyama, Apr 04 2021

sqrt(2+sqrt(2+sqrt(2)))/2 \\ Seiichi Manyama, Apr 04 2021

Equals (1/2) * sqrt(2+sqrt(2+sqrt(2))). - Seiichi Manyama, Apr 04 2021
Root of 128*x^8 -256*x^6 +160*x^4 -32*x^2 +1 = 0. - R. J. Mathar, Aug 29 2025
2*this^2 -1 = A144981. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/8,1/8;1/2;1/2). - R. J. Mathar, Aug 31 2025

A019907 Decimal expansion of tangent of 9 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the decimal expansion of cotangent of 81 degrees. - Mohammad K. Azarian, Jun 30 2013

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

RealDigits[Tan[9 Degree],10,120][[1]] (* Harvey P. Dale, Aug 23 2020 *)

Equals tan(Pi/20) = A019818/A019890. - R. J. Mathar, Aug 29 2025

Smallest positive of the 8 real-valued roots of x^8-44*x^6+166*x^4-44*x^2+1=0. (Others A019925, A019961, A019979). - R. J. Mathar, Aug 31 2025

A343056 Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.9951847266721968862448369...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Leon D. Fairbanks, Powers of Cosine and Sine, arXiv:2308.04437 [math.GM], 2023. See p. 3.
Wikipedia, Trigonometric constants expressed in real radicals.
Index entries for algebraic numbers, degree 16

cos(Pi/m): A010503 (m=4), A019863 (m=5), A010527 (m=6), A073052 (m=7), A144981 (m=8), A019879 (m=9), A019881 (m=10), A019884 (m=12), A232735 (m=14), A019887 (m=15), A232737 (m=16), A210649 (m=17), A019889 (m=18), A019890 (m=20), A144982 (m=24), A019893 (m=30). this sequence (m=32), A019894 (m=36).

R:= RealField(127); Cos(Pi(R)/32); // G. C. Greubel, Sep 30 2022

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

PARI
```
real(I^(1/16))
```
PARI
```
cos(Pi/32)
```
PARI
```
sqrt(2+sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(cos(pi/32), digits=122) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2+sqrt(2+sqrt(2+sqrt(2)))).
Satisfies 32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +84480*x^8 -21504*x^6 +2688*x^4 -128*x^2 +1 = 0. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/16,1/16;1/2;1/2). - R. J. Mathar, Aug 31 2025

cp's OEIS Frontend

A019827 Decimal expansion of sin(Pi/10) (angle of 18 degrees).

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A019818 Decimal expansion of sine of 9 degrees.

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A232735 Decimal expansion of the real part of I^(1/7), or cos(Pi/14).

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A019836 Decimal expansion of sine of 27 degrees.

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A019872 Decimal expansion of sine of 63 degrees.

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A232737 Decimal expansion of the real part of I^(1/8), or cos(Pi/16).

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A019907 Decimal expansion of tangent of 9 degrees.

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