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%I #9 Jul 01 2013 10:35:38
%S 2,9,7,1,0,5,6,3,5,2,7,4,8,2,2,7,1,6,7,1,2,2,1,4,4,3,6,5,2,6,3,1,6,1,
%T 9,9,4,0,7,2,9,6,0,7,1,0,8,5,6,7,0,4,0,0,5,6,7,6,8,6,4,5,5,2,4,8,5,8,
%U 2,3,6,9,4,8,4,1,8,0,8,1,7,7,0,0,6,8,2,3,8,4,1,4,6,4,9,0,9,4,3
%N Decimal expansion of the radius of the circle tangent to the curve y=cos(2x) and to the positive x and y axes.
%C Let (x,y) denote the point of tangency. Then
%C x=0.556627409764774263651183045638839616840052780212...
%C y=0.441743828977740325730277185387438343947805907493...
%C slope=-0.5283257380737094443139057566841614427843590...
%C (The Mathematica program includes a graph.)
%e radius=0.2971056352748227167122144365263161994072960710...
%t r = .297; c = 2;
%t Show[Plot[Cos[c*x], {x, 0, Pi}],
%t ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}],PlotRange -> All, AspectRatio -> Automatic]
%t f[x_] := (x - c*Sin[c*x] Cos[c*x])/(1 - c*Sin[c*x]);
%t t = x /. FindRoot[Cos[c*x] == f[x] + Sqrt[2*f[x]*x - x^2], {x, .5, 1}, WorkingPrecision -> 100]
%t x1 = Re[t] (* x coordinate of tangency point *)
%t y = Cos[c*x1] (* y coordinate of tangency point *)
%t radius = f[x1]
%t RealDigits[radius] (* A197017 *)
%t slope = -Sin[x1] (* slope at tangency point *)
%Y Cf. A197016, A197018, A197019, A197020.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Oct 08 2011
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