A197027 Decimal expansion of the radius of the smallest circle tangent to the x axis and to the curve y=-cos(2x) at points (x,y), (-x,y).

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

Offset: 1

Author

Clark Kimberling, Oct 09 2011

Keywords

• nonn
• cons

Comments

Let (x,y) denote the point of tangency, where x>0:

x=1.116022083263345851737313595257930429156...

y=0.614102158068589478230317674621719242848...

slope=1.578453090155676616014698886314279426...

(The Mathematica program includes a graph.)

Examples

			radius=1.321137476522859782809009849582516244...
		

Crossrefs

Cf. A197026, A196028, A196021.

Programs

Mathematica

r = 1.32; c = 2;
Show[Plot[-Cos[c*x], {x, -4, 4}],
 ContourPlot[x^2 + (y - r)^2 == r^2, {x, -3, 3}, {y, -1.5, 3}], PlotRange -> All, AspectRatio -> Automatic]
u[x_] := -Cos[c*x] + x/(c*Sin[c*x]);
t1 = x /. FindRoot[Sqrt[u[x]^2 - x^2] == u[x] + Cos[c*x], {x, 1, 1.5}, WorkingPrecision -> 100]
t = Re[t1];
RealDigits[t]   (* x coordinate of tangency point *)
y = -Cos[c*t]   (* y coordinate of tangency point *)
radius = u[t]
RealDigits[radius] (* A197027 *)
slope = c*Sin[c*t] (* slope at tangency point *)