A197028 Decimal expansion of the radius of the smallest circle tangent to the x axis and to the curve y=-cos(3x) at points (x,y), (-x,y).

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

Offset: 0

Author

Clark Kimberling, Oct 09 2011

Keywords

• nonn
• cons

Comments

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

x=0.6888117352645178597708892254141829843113...

y=0.4755937478149254230061087613442876576146...

slope=2.6389951275730271940627334805152084806...

(The Mathematica program includes a graph.)

Examples

			radius=0.7366066341471518249920789050824520648...
		

Crossrefs

Cf. A197026, A196027, A196029, A196021.

Programs

Mathematica

r = .737; c = 3;
Show[Plot[-Cos[c*x], {x, -2, 2}],
 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, .6, .8}, 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] (* A197028 *)
slope = c*Sin[c*t] (* slope at tangency point *)