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A323601 Decimal expansion of sin(Pi/7).

%I A323601 #33 Aug 31 2025 09:00:56
%S A323601 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 A323601 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 A323601 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 A323601 Decimal expansion of sin(Pi/7).
%H A323601 G. C. Greubel, <a href="/A323601/b323601.txt">Table of n, a(n) for n = 0..10000</a>
%H A323601 <a href="/index/Al#algebraic_06">Index entries for algebraic numbers, degree 6</a>.
%F A323601 Root of the equation 64*x^6 - 112*x^4 + 56*x^2 - 7 = 0. (Other +- A232735 and +- 0.7818314... = +- cos(3*Pi/14))
%F A323601 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 A323601 Equals cos(5*Pi/14).
%F A323601 From _Gleb Koloskov_, Jul 15 2021: (Start)
%F A323601 Positive root of the equation x^3 + sqrt(7)/2*x^2 - sqrt(7)/8 = 0.
%F A323601 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)
%F A323601 Equals 1/A121598 = A272487/2. - _Hugo Pfoertner_, Dec 15 2024
%F A323601 This^2 + A073052^2=1. - _R. J. Mathar_, Aug 31 2025
%e A323601 0.43388373911755812047576833284835875460999072778745987644454730353220325...
%t A323601 RealDigits[Sin[Pi/7], 10, 120][[1]]
%o A323601 (PARI) default(realprecision, 100); sin(Pi/7) \\ _G. C. Greubel_, Feb 08 2019
%o A323601 (PARI) polrootsreal(64*x^6-112*x^4+56*x^2-7)[4] \\ _Charles R Greathouse IV_, Feb 05 2025
%o A323601 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Sin(Pi(R)/7); // _G. C. Greubel_, Feb 08 2019
%o A323601 (Sage) numerical_approx(sin(pi/7), digits=100) # _G. C. Greubel_, Feb 08 2019
%Y A323601 Cf. A019829 (sin(Pi/9)), A232736 (sin(Pi/14)).
%Y A323601 Cf. A121598, A272487.
%K A323601 nonn,cons,changed
%O A323601 0,1
%A A323601 _Vaclav Kotesovec_, Jan 19 2019