A197016 - OEIS

%I #10 Jul 01 2013 10:35:49

%S 4,2,8,7,7,8,5,3,6,0,3,0,6,1,2,8,6,3,6,1,3,6,9,1,7,4,1,0,4,8,9,9,9,7,

%T 0,4,9,0,6,0,5,8,9,3,6,1,5,2,0,2,6,8,5,1,9,9,3,7,8,8,2,4,6,9,8,4,7,1,

%U 3,9,3,2,2,8,8,8,9,7,9,4,8,6,0,3,5,1,0,1,5,5,4,3,3,2,3,1,2,3,6

%N Decimal expansion of the radius of the circle tangent to the curve y=cos(x) and to the positive x and y axes.

%C Let (x,y) denote the point of tangency.  Then

%C x=0.65099256993050253383380179140902527170294599...

%C y=0.79548271667012269646991174255794794798663548...

%C slope=-0.6059762763335882427824587356062000...

%C (The Mathematica program includes a graph.)

%e radius=0.428778536030612863613691741048999...

%t r = .428;

%t Show[Plot[Cos[x], {x, 0, Pi}],

%t ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]

%t f[x_] := (x - Sin[x] Cos[x])/(1 - Sin[x]);

%t t = x /.FindRoot[Cos[x] == f[x] + Sqrt[2*f[x]*x - x^2], {x, .5, 1}, WorkingPrecision -> 100]

%t x1 = Re[t]  (* x coordinate of tangency point *)

%t y = Cos[x1] (* y coordinate of tangency point *)

%t radius = f[x1]

%t RealDigits[radius] (* A197016 *)

%t slope = -Sin[x1]   (* slope at tangency point *)

%Y Cf. A197017, A197018, A197019, A197020.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Oct 08 2011