A323601 Decimal expansion of sin(Pi/7).

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

Author

Vaclav Kotesovec, Jan 19 2019

Keywords

• nonn
• cons
• changed

Examples

			0.43388373911755812047576833284835875460999072778745987644454730353220325...
		

Links

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

Crossrefs

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

Cf. A121598, A272487.

Programs

Magma

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

Mathematica

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

PARI

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

PARI

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

Sage

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

Formula

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