%I #8 Mar 07 2013 09:09:34
%S 3,0,4,6,7,5,3,6,3,3,0,6,6,0,7,4,5,2,4,0,2,1,6,8,4,3,1,6,6,7,7,5,8,1,
%T 9,5,4,8,5,6,3,6,3,6,9,6,8,1,4,2,8,7,7,9,1,0,1,1,4,9,8,1,8,4,3,6,2,9,
%U 3,8,8,3,2,7,2,6,0,2,1,7,2,3,5,2,6,2,5,4,5,3,2,3,4,0,4,7,2,7,8
%N Decimal expansion of the radius of the circle tangent to the curve y=cos(4x) at points (x,y) and (-x,y), where 0<x<1.
%C Let (x,y) denote the point of tangency. Then
%C x=0.294083445311344461181635110698988639348667...
%C y=0.384312064643508105468613486692501669417807...
%C slope=-3.69281299167871547859350850472131295652...
%C (The Mathematica program includes a graph.)
%e radius=0.30467536330660745240216843166775819548...
%t r = .304; c = 4;
%t Show[Plot[Cos[c*x], {x, -.5, .5}],
%t ContourPlot[x^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
%t t = x /. FindRoot[
%t c*Sin[c*x] Cos[c*x] - x == x*Sqrt[1 + (c*Sin[c*x])^2], {x, .25, .55}, WorkingPrecision -> 100]
%t RealDigits[t] (* x coordinate of tangency point *)
%t y = Cos[c*t] (* y coordinate of tangency point *)
%t radius = Cos[c*t] - t/(c*Sin[c*t])
%t RealDigits[radius] (* A197022 *)
%t slope = -c*Sin[c*t] (* slope at tangency point *)
%Y Cf. A197020, A196021, A196026, A197027, A197016.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Oct 08 2011