A232735 -id:A232735 - cp's OEIS frontend

A232736 Decimal expansion of sin(Pi/14), or the imaginary part of (-1)^(1/7).

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding real part is in A232735.
Root of the equation 1 - 4*x - 4*x^2 + 8*x^3 = 0. - Vaclav Kotesovec, Apr 04 2021
The other 2 roots are -A362922 and A073052. - R. J. Mathar, Aug 29 2025

			0.222520933956314404288902564496794759466355568764544955311987...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 3.

Cf. A232735 (real part), A010503 (imag(I^(1/2))), A182168 (imag(I^(1/4))), A019827 (imag(I^(1/5))), A019824 (imag(I^(1/6))), A232738 (imag(I^(1/8))), A019819 (imag(I^(1/9))), A019818 (imag(I^(1/10))).

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

A323601 Decimal expansion of sin(Pi/7).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jan 19 2019

			0.43388373911755812047576833284835875460999072778745987644454730353220325...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for algebraic numbers, degree 6.

Cf. A019829 (sin(Pi/9)), A232736 (sin(Pi/14)).

Cf. A121598, A272487.

SetDefaultRealField(RealField(100)); R:= RealField(); Sin(Pi(R)/7); // G. C. Greubel, Feb 08 2019

Mathematica
```
RealDigits[Sin[Pi/7], 10, 120][[1]]
```

default(realprecision, 100); sin(Pi/7) \\ G. C. Greubel, Feb 08 2019

polrootsreal(64*x^6-112*x^4+56*x^2-7)[4] \\ Charles R Greathouse IV, Feb 05 2025

numerical_approx(sin(pi/7), digits=100) # G. C. Greubel, Feb 08 2019

Root of the equation 64*x^6 - 112*x^4 + 56*x^2 - 7 = 0. (Other +- A232735 and +- 0.7818314... = +- cos(3*Pi/14))
Equals sqrt((196 + 7*i*2^(2/3)*(21*i*sqrt(3) - 7)^(1/3)*(i + sqrt(3)) + i*2^(4/3)*(21*i*sqrt(3) - 7)^(2/3)*(2*i + sqrt(3)))/336), where i is the imaginary unit.
Equals cos(5*Pi/14).
From Gleb Koloskov, Jul 15 2021: (Start)
Positive root of the equation x^3 + sqrt(7)/2*x^2 - sqrt(7)/8 = 0.
Equals ((4*sqrt(7)*(13+3*sqrt(3)*i))^(1/3)+28*(4*sqrt(7)*(13+3*sqrt(3)*i))^(-1/3)-2*sqrt(7))/12, where i is the imaginary unit. (End)
Equals 1/A121598 = A272487/2. - Hugo Pfoertner, Dec 15 2024
This^2 + A073052^2=1. - R. J. Mathar, Aug 31 2025

A343059 Decimal expansion of tan(Pi/14).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

Root of the equation -1 + 21*x^2 - 35*x^4 + 7*x^6 = 0. - Vaclav Kotesovec, Apr 04 2021

			0.228243474390149938077611362061014782...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for algebraic numbers, degree 6.

Cf. A232736 (sin(Pi/14)), A232735 (cos(Pi/14)).

RealDigits[Tan[Pi/14], 10, 125][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
tan(Pi/14)
```

numerical_approx(tan(pi/14), digits=124) # G. C. Greubel, Sep 30 2022

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

A387448 Decimal expansion of cos(Pi/28).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 29 2025

			0.99371220989324258353...

R. J. Mathar, Reduction formulas of the cosine of integer fractions of pi, (2008) vixra:2210.0026
Index entries for algebraic numbers, degree 12

Cf. A132744.

Equals sin(13*Pi/28) = sqrt( (A232735+1)/2 ) = 2F1(-1/14,1/14;1/2;1/2).

Largest of the 12 real-valued roots of 4096*x^12 -12288*x^10 +13568*x^8 -6656*x^6 +1376*x^4 -96*x^2 +1 =0.

cp's OEIS Frontend

A232736 Decimal expansion of sin(Pi/14), or the imaginary part of (-1)^(1/7).

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

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A323601 Decimal expansion of sin(Pi/7).

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A343059 Decimal expansion of tan(Pi/14).

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

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A387448 Decimal expansion of cos(Pi/28).

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