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A197022 Decimal expansion of the radius of the circle tangent to the curve y=cos(4x) at points (x,y) and (-x,y), where 0

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