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%I #5 Mar 30 2012 18:57:52
%S 7,3,6,6,0,6,6,3,4,1,4,7,1,5,1,8,2,4,9,9,2,0,7,8,9,0,5,0,8,2,4,5,2,0,
%T 6,4,8,2,2,7,6,0,6,3,9,9,8,3,9,0,2,7,9,1,5,0,8,1,4,8,0,8,0,6,8,3,6,8,
%U 0,1,0,5,1,2,3,8,5,3,9,8,9,0,6,3,9,4,3,6,5,7,3,0,8,0,0,9,2,6,2
%N Decimal expansion of the radius of the smallest circle tangent to the x axis and to the curve y=-cos(3x) at points (x,y), (-x,y).
%C Let (x,y) denote the point of tangency, where x>0:
%C x=0.6888117352645178597708892254141829843113...
%C y=0.4755937478149254230061087613442876576146...
%C slope=2.6389951275730271940627334805152084806...
%C (The Mathematica program includes a graph.)
%e radius=0.7366066341471518249920789050824520648...
%t r = .737; c = 3;
%t Show[Plot[-Cos[c*x], {x, -2, 2}],
%t ContourPlot[x^2 + (y - r)^2 == r^2, {x, -3, 3}, {y, -1.5, 3}], PlotRange -> All, AspectRatio -> Automatic]
%t u[x_] := -Cos[c*x] + x/(c*Sin[c*x]);
%t t1 = x /. FindRoot[Sqrt[u[x]^2 - x^2] == u[x] + Cos[c*x], {x, .6, .8}, WorkingPrecision -> 100]
%t t = Re[t1];
%t RealDigits[t] (* x coordinate of tangency point *)
%t y = -Cos[c*t] (* y coordinate of tangency point *)
%t radius = u[t]
%t RealDigits[radius] (* A197028 *)
%t slope = c*Sin[c*t] (* slope at tangency point *)
%Y Cf. A197026, A196027, A196029, A196021.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Oct 09 2011
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