A197029 Decimal expansion of the radius of the smallest circle tangent to the x axis and to the curve y=-cos(4x) at points (x,y), (-x,y).

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

Offset: 0

Author

Clark Kimberling, Oct 09 2011

Keywords

• nonn
• cons

Comments

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

x=0.488618197079923270050681129865078039260837...

y=0.374332154777652501331094642913853652491893...

slope=3.709178750935618333987343550424591912283...

(The Mathematica program includes a graph.)

Examples

			radius=0.5060643332165245100546376217734714411...
		

Crossrefs

Cf. A197026, A196027, A196028, A196022.

Programs

Mathematica

r = .5; c = 4;
Show[Plot[-Cos[c*x], {x, -1, 1}],
 ContourPlot[x^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1.5, 2}], 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, .4, .5}, WorkingPrecision -> 100]
t = Re[t1]    (* x coordinate of tangency point *)
y = -Cos[c*t] (* y coordinate of tangency point *)
radius = u[t]
RealDigits[radius] (* A197029 *)
slope = c*Sin[c*t] (* slope at tangency point *)