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A195703 Decimal expansion of arccos(sqrt(3/8)) and of arcsin(sqrt(5/8)).

%I A195703 #21 Nov 21 2024 07:40:47
%S A195703 9,1,1,7,3,8,2,9,0,9,6,8,4,8,7,6,3,6,3,5,8,4,8,9,5,6,4,3,1,6,7,3,1,2,
%T A195703 0,7,1,7,5,3,8,9,2,1,6,3,9,2,1,9,5,5,5,2,0,6,0,6,9,8,0,3,7,4,4,7,2,4,
%U A195703 1,6,3,1,8,1,2,0,6,2,8,6,0,8,6,2,8,8,3,0,7,7,4,4,9,5,3,6,5,6,7,7
%N A195703 Decimal expansion of arccos(sqrt(3/8)) and of arcsin(sqrt(5/8)).
%C A195703 From _Roman Witula_, Oct 31 2012: (Start)
%C A195703 It can be deduced that cos(a)*cos(3*a) = min {cos(x)*cos(3*x): x in R} = -9/16 = -0,5625. Moreover we have 1 = cos(Pi)*cos(3*Pi) = max {cos(x)*cos(3*x): x in R}.
%C A195703 Note that also cot(3*a) = -3*cot(a) since d(-cos(x)*cos(3*x))/dx = sin(x)*cos(3*x) + 3*cos(x)*sin(3*x). Moreover we have max{F(x): x in R} = -2*sqrt(2)*cos(a)*cos(3*a) = 9*sqrt(2)/8, where F(x) := cos(x) + sin(x) - sqrt(2)*sin(2*x). Indeed, we obtain F(x) = sqrt(2)*cos(x)*(sin(Pi/4) - sin(x)) + sqrt(2)*sin(x)*(cos(Pi/4) - cos(x)) = 2*sqrt(2)*sin(Pi/8 - x/2)*(cos(x)*cos(Pi/8 + x/2) - sin(x)*sin(Pi/8 + x/2)) = 2*sqrt(2)*sin(Pi/8 - x/2)*cos(3*x/2 + Pi/8) = -2*sqrt(2)*cos((x - 5*Pi/4)/2)*cos(3*(x - 5*Pi/4)/2).
%C A195703 (End)
%H A195703 G. C. Greubel, <a href="/A195703/b195703.txt">Table of n, a(n) for n = 0..5000</a>
%H A195703 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A195703 Equals arctan(sqrt(5/3)). - _Amiram Eldar_, Jul 09 2023
%e A195703 arccos(sqrt(3/8)) = 0.91173829096848763635848956431...
%t A195703 r = Sqrt[3/8];
%t A195703 N[ArcSin[r], 100]
%t A195703 RealDigits[%]  (* A195700 *)
%t A195703 N[ArcCos[r], 100]
%t A195703 RealDigits[%]  (* A195703 *)
%t A195703 N[ArcTan[r], 100]
%t A195703 RealDigits[%]  (* A195705 *)
%t A195703 N[ArcCos[-r], 100]
%t A195703 RealDigits[%]  (* A195706 *)
%o A195703 (PARI) acos(sqrt(3/8)) \\ _G. C. Greubel_, Nov 18 2017
%o A195703 (Magma) [Arccos(Sqrt(3/8))]; // _G. C. Greubel_, Nov 18 2017
%Y A195703 Cf. A195700.
%K A195703 nonn,cons
%O A195703 0,1
%A A195703 _Clark Kimberling_, Sep 23 2011