A197021 Decimal expansion of the radius of the circle tangent to the curve y=cos(3x) at points (x,y) and (-x,y), where 0

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

Offset: 0

Author

Clark Kimberling, Oct 08 2011

Keywords

• nonn
• cons

Comments

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

x=0.346818914654599529577486037538498433565584415464...

y=0.505826306518745297430716717373078359704411629139...

slope=-2.5879060509806840663013781941932136174746999...

(The Mathematica program includes a graph.)

Examples

			radius=0.3718110417361721840195647351588579028970626...
		

Crossrefs

Cf. A197020, A196022, A196026, A197027, A197016.

Programs

Mathematica

r = .371; c = 3;
Show[Plot[Cos[c*x], {x, -0.5, 0.5}],
 ContourPlot[x^2 + (y - r)^2 == r^2, {x, -1, 1}, {y, -1, 1}], PlotRange -> All, AspectRatio -> Automatic]
t = x /. FindRoot[c*Sin[c*x] Cos[c*x] - x == x*Sqrt[1 + (c*Sin[c*x])^2], {x, .25, .55}, WorkingPrecision -> 100]
RealDigits[t]   (* x coordinate of tangency point *)
y = Cos[c*t]    (* y coordinate of tangency point *)
radius = Cos[c*t] - t/(c*Sin[c*t])
RealDigits[radius]   (* A197021 *)
slope = -c*Sin[c*t]  (* slope at tangency point *)