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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.
1

%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