A197019 Decimal expansion of the radius of the circle tangent to the curve y=cos(4x) and to the positive x and y axes.

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

Offset: 0

Author

Clark Kimberling, Oct 08 2011

Keywords

• nonn
• cons

Comments

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

x=0.33861718723736417045737960551501765846156681578...

y=0.21464425212782002883052365316387247038020190838...

slope=-0.332183120530610097233795968342303024088179...

(The Mathematica program includes a graph.)

Examples

			radius=0.171994517348101690739024865448714954394...
		

Crossrefs

Cf. A197016, A196017, A196018, A197020.

Programs

Mathematica

r = .172; c = 4;
Show[Plot[Cos[c*x], {x, 0, Pi}],
 ContourPlot[(x - r)^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
f[x_] := (x - c*Sin[c*x] Cos[c*x])/(1 - c*Sin[c*x]);
t = x /. FindRoot[Cos[c*x] == f[x] + Sqrt[2*f[x]*x - x^2], {x, .5, 1}, WorkingPrecision -> 100]
x1 = Re[t]    (* x coordinate of tangency point *)
y = Cos[c*x1] (* y coordinate of tangency point *)
radius = f[x1]
RealDigits[radius] (* A197019 *)
slope = -Sin[x1]   (* slope at tangency point *)