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%I #38 Dec 22 2023 14:19:11
%S 4,0,3,9,7,2,7,5,3,2,9,9,5,1,7,2,0,9,3,1,8,9,6,1,7,4,0,0,6,6,3,1,5,4,
%T 4,2,9,0,2,2,3,5,9,6,4,5,7,4,0,9,8,4,2,2,2,5,0,0,9,7,6,0,1,7,3,3,8,7,
%U 0,5,4,9,9,7,1,2,9,5,3,5,3,5,0,1,2,4,3,3,9,0,1,6,5,2,2,2,7,2,8,7,0,9,4,9,1
%N Decimal expansion of the solution to x*sqrt(1-x^2) + arcsin(x) = Pi/4, or the length of the line connecting the origin to the center of the chord of a circle, centered at 0 and of radius 1, that divides the circle such that 1/4 of the area is on one side and 3/4 is on the other side.
%C Decimal expansion of the number sin(u'), where u' is the number in [0,2 Pi] such that the line normal to the graph of y = sin x at (u', sin u') passes through the point (3 Pi/4,0). See A332500. - _Clark Kimberling_, May 05 2020
%H Robert P. P. McKone, <a href="/A086751/b086751.txt">Table of n, a(n) for n = 0..19999</a>
%F Define k(n+1) as k(n) - (k(n)*sqrt(1-k(n)^2) + arcsin(k(n)) - Pi/4). The sequence is the decimal expansion of lim_{n -> infinity} k(n).
%F Equals sqrt(2-2*sqrt(1-d^2))/2, where d = A003957 is the Dottie number. - _Gleb Koloskov_, Jun 16 2021
%e 0.403972753299517...
%p Digits := 240 ; x := 0.4 ; for i from 1 to 8 do f := sin(2.0*x)+2.0*x-Pi/2.0 ; fp := 2*cos(2*x)+2.0 ; x := x-evalf(f/fp) ; printf("%.120f\n",sin(x)) ; od: x := sin(x) ; read("transforms3") ; CONSTTOLIST(x) ; # _R. J. Mathar_, May 19 2009
%t digits = 105; Sin[FindRoot[Sin[2*a]/2+a == Pi/4, {a, 1/2}, WorkingPrecision -> digits][[1, 2]]] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 21 2014 *)
%o (PARI) solve(x=0, 1, x*sqrt(1-x^2) + asin(x) - Pi/4) \\ _Michel Marcus_, May 05 2020
%o (PARI) my(d=solve(x=0,1,cos(x)-x)); sqrt(2-2*sqrt(1-d^2))/2 \\ _Gleb Koloskov_, Jun 16 2021
%Y Cf. A003957.
%K cons,nonn,easy
%O 0,1
%A Jonathan R. Anderson (neo__jon(AT)hotmail.com), Jul 30 2003
%E More terms from _Jim Nastos_, Sep 05 2003
%E More digits from _R. J. Mathar_, May 19 2009
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