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A195699 Decimal expansion of arcsin(sqrt(1/8)) and of arccos(sqrt(7/8)).

%I A195699 #19 Nov 21 2024 07:40:49
%S A195699 3,6,1,3,6,7,1,2,3,9,0,6,7,0,7,8,0,5,5,8,9,1,8,8,6,7,6,3,2,0,6,6,6,6,
%T A195699 8,1,0,1,2,6,0,9,2,4,3,2,1,2,2,2,0,1,3,3,8,1,3,3,7,7,0,6,6,2,9,1,8,5,
%U A195699 3,6,9,0,9,5,7,3,1,5,1,3,2,4,8,2,4,1,3,8,0,5,4,6,9,5,5,0,6,5,1,8
%N A195699 Decimal expansion of arcsin(sqrt(1/8)) and of arccos(sqrt(7/8)).
%H A195699 G. C. Greubel, <a href="/A195699/b195699.txt">Table of n, a(n) for n = 0..5000</a>
%H A195699 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A195699 From _Peter Bala_, Jan 14 2022: (Start)
%F A195699 Equals (1/2)*arccos(3/4) = arctan(sqrt(7)/7).
%F A195699 Equals sqrt(7)*Sum_{n >= 0} 1/((16*n + 8)*(2^n)*binomial(2*n,n)).
%F A195699 Equals sqrt(2)*Sum_{n >= 0} binomial(2*n,n)/((8*n + 4)*32^n). (End)
%e A195699 arcsin(sqrt(1/8)) = 0.3613671239067078055891886763206666...
%t A195699 r = Sqrt[1/8];
%t A195699 N[ArcSin[r], 100]
%t A195699 RealDigits[%]  (* A195699 *)
%t A195699 N[ArcCos[r], 100]
%t A195699 RealDigits[%]  (* A168229 *)
%t A195699 N[ArcTan[r], 100]
%t A195699 RealDigits[%]  (* A188615 *)
%t A195699 N[ArcCos[-r], 100]
%t A195699 RealDigits[%]  (* A195704 *)
%o A195699 (PARI) asin(sqrt(1/8)) \\ _G. C. Greubel_, Nov 18 2017
%o A195699 (Magma) [Arcsin(Sqrt(1/8))]; // _G. C. Greubel_, Nov 18 2017
%Y A195699 Cf. A195700, A195703, A195704.
%K A195699 nonn,cons
%O A195699 0,1
%A A195699 _Clark Kimberling_, Sep 23 2011