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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).
5

%I #5 Mar 30 2012 18:57:52

%S 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 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 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 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 Let (x,y) denote the point of tangency, where x>0:

%C x=1.116022083263345851737313595257930429156...

%C y=0.614102158068589478230317674621719242848...

%C slope=1.578453090155676616014698886314279426...

%C (The Mathematica program includes a graph.)

%e radius=1.321137476522859782809009849582516244...

%t r = 1.32; c = 2;

%t Show[Plot[-Cos[c*x], {x, -4, 4}],

%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, 1, 1.5}, 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] (* A197027 *)

%t slope = c*Sin[c*t] (* slope at tangency point *)

%Y Cf. A197026, A196028, A196021.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Oct 09 2011