A197026 Decimal expansion of the radius of the smallest circle tangent to the x axis and to the curve y=-cos(x) at points (x,y), (-x,y).

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

Offset: 1

Author

Clark Kimberling, Oct 09 2011

Keywords

• nonn
• cons

Comments

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

x=2.3973091169572703557415944811143634671454653692...

y=0.7355734556385944841653303915319993812641279844...

slope=0.6774449729386857532010706302057868510403567...

(The Mathematica program includes a graph.)

Examples

			radius=4.27432451693585827192680241796164720368009482...
		

Crossrefs

Cf. A197027, A196028.

Programs

Mathematica

r = 4.27; c = 1;
Show[Plot[-Cos[c*x], {x, -5, 5}],
ContourPlot[x^2 + (y - r)^2 == r^2, {x, -5, 5}, {y, -1.5, 8.7}], PlotRange -> All, AspectRatio -> Automatic]
u[x_] := -Cos[c*x] + x/(c*Sin[c*x]);
t = x /. FindRoot[Sqrt[u[x]^2 - x^2] == u[x] + Cos[c*x], {x, 2, 3}, WorkingPrecision -> 100]
RealDigits[t]   (* x coordinate of tangency point *)
y = -Cos[c*t]   (* y coordinate of tangency point *)
radius = u[t]
RealDigits[radius] (* A197026 *)
slope = c*Sin[c*t]   (* slope at tangency point *)