%I #20 Sep 08 2022 08:46:23

%S 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,

%T 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,

%U 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

%N Decimal expansion of sin(Pi/7).

%H G. C. Greubel, <a href="/A323601/b323601.txt">Table of n, a(n) for n = 0..10000</a>

%F Root of the equation 64*x^6 - 112*x^4 + 56*x^2 - 7 = 0.

%F 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.

%F Equals cos(5*Pi/14).

%F From _Gleb Koloskov_, Jul 15 2021: (Start)

%F Positive root of the equation x^3 + sqrt(7)/2*x^2 - sqrt(7)/8 = 0.

%F 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)

%e 0.43388373911755812047576833284835875460999072778745987644454730353220325...

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

%o (PARI) default(realprecision, 100); sin(Pi/7) \\ _G. C. Greubel_, Feb 08 2019

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Sin(Pi(R)/7); // _G. C. Greubel_, Feb 08 2019

%o (Sage) numerical_approx(sin(pi/7), digits=100) # _G. C. Greubel_, Feb 08 2019

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

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, Jan 19 2019