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A197026 Decimal expansion of the radius of the smallest circle tangent to the x axis and to the curve y=-cos(x) at points (x,y), (-x,y).

%I A197026 #8 Feb 02 2013 13:00:21
%S A197026 4,2,7,4,3,2,4,5,1,6,9,3,5,8,5,8,2,7,1,9,2,6,8,0,2,4,1,7,9,6,1,6,4,7,
%T A197026 2,0,3,6,8,0,0,9,4,8,2,8,2,9,0,5,0,9,5,2,2,1,7,3,9,7,4,6,3,4,1,8,3,1,
%U A197026 9,9,8,4,8,5,3,6,3,3,8,1,6,4,3,8,6,8,1,5,0,4,5,9,7,8,4,7,7,6,1
%N A197026 Decimal expansion of the radius of the smallest circle tangent to the x axis and to the curve y=-cos(x) at points (x,y), (-x,y).
%C A197026 Let (x,y) denote the point of tangency, where x>0:
%C A197026 x=2.3973091169572703557415944811143634671454653692...
%C A197026 y=0.7355734556385944841653303915319993812641279844...
%C A197026 slope=0.6774449729386857532010706302057868510403567...
%C A197026 (The Mathematica program includes a graph.)
%e A197026 radius=4.27432451693585827192680241796164720368009482...
%t A197026 r = 4.27; c = 1;
%t A197026 Show[Plot[-Cos[c*x], {x, -5, 5}],
%t A197026 ContourPlot[x^2 + (y - r)^2 == r^2, {x, -5, 5}, {y, -1.5, 8.7}], PlotRange -> All, AspectRatio -> Automatic]
%t A197026 u[x_] := -Cos[c*x] + x/(c*Sin[c*x]);
%t A197026 t = x /. FindRoot[Sqrt[u[x]^2 - x^2] == u[x] + Cos[c*x], {x, 2, 3}, WorkingPrecision -> 100]
%t A197026 RealDigits[t]   (* x coordinate of tangency point *)
%t A197026 y = -Cos[c*t]   (* y coordinate of tangency point *)
%t A197026 radius = u[t]
%t A197026 RealDigits[radius] (* A197026 *)
%t A197026 slope = c*Sin[c*t]   (* slope at tangency point *)
%Y A197026 Cf. A197027, A196028.
%K A197026 nonn,cons
%O A197026 1,1
%A A197026 _Clark Kimberling_, Oct 09 2011