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A197027 Decimal expansion of the radius of the smallest circle tangent to the x axis and to the curve y=-cos(2x) at points (x,y), (-x,y).

%I A197027 #5 Mar 30 2012 18:57:52
%S A197027 1,3,2,1,1,3,7,4,7,6,5,2,2,8,5,9,7,8,2,8,0,9,0,0,9,8,4,9,5,8,2,5,1,6,
%T A197027 2,4,4,3,1,5,6,3,7,9,7,6,8,2,7,5,4,6,2,6,4,4,3,4,6,5,0,4,2,9,9,8,3,5,
%U A197027 8,8,3,0,0,9,6,6,5,9,9,7,7,4,3,6,5,9,4,4,1,1,3,4,6,0,4,5,4,3,9,9
%N A197027 Decimal expansion of the radius of the smallest circle tangent to the x axis and to the curve y=-cos(2x) at points (x,y), (-x,y).
%C A197027 Let (x,y) denote the point of tangency, where x>0:
%C A197027 x=1.116022083263345851737313595257930429156...
%C A197027 y=0.614102158068589478230317674621719242848...
%C A197027 slope=1.578453090155676616014698886314279426...
%C A197027 (The Mathematica program includes a graph.)
%e A197027 radius=1.321137476522859782809009849582516244...
%t A197027 r = 1.32; c = 2;
%t A197027 Show[Plot[-Cos[c*x], {x, -4, 4}],
%t A197027  ContourPlot[x^2 + (y - r)^2 == r^2, {x, -3, 3}, {y, -1.5, 3}], PlotRange -> All, AspectRatio -> Automatic]
%t A197027 u[x_] := -Cos[c*x] + x/(c*Sin[c*x]);
%t A197027 t1 = x /. FindRoot[Sqrt[u[x]^2 - x^2] == u[x] + Cos[c*x], {x, 1, 1.5}, WorkingPrecision -> 100]
%t A197027 t = Re[t1];
%t A197027 RealDigits[t]   (* x coordinate of tangency point *)
%t A197027 y = -Cos[c*t]   (* y coordinate of tangency point *)
%t A197027 radius = u[t]
%t A197027 RealDigits[radius] (* A197027 *)
%t A197027 slope = c*Sin[c*t] (* slope at tangency point *)
%Y A197027 Cf. A197026, A196028, A196021.
%K A197027 nonn,cons
%O A197027 1,2
%A A197027 _Clark Kimberling_, Oct 09 2011