A196996 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) at which a line y=m*x meets the curve y=cos(3x) orthogonally.

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

Offset: 0

Author

Clark Kimberling, Oct 09 2011

Keywords

• nonn
• cons

Comments

See the Mathematica program for a graph.

xo=0.9350272884749678361451944275323...

yo=0.3301955980451199836007253971727...

m=0.35314006565912096755666111412785...

|OP|=0.99161744799152518925689622748...

Crossrefs

Cf. A196997, A197000, A197002.

Programs

Mathematica

c = 3;
xo = x /. FindRoot[0 == x + c*Sin[c*x] Cos[c*x], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A196996 *)
m = Sin[c*xo]/xo
RealDigits[m]  (* A196997 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Sin[c*x], yo - (1/m) (x - xo)}, {x, 0, Pi/c}], ContourPlot[{y == m*x}, {x, 0, 1.5}, {y, -.1, 1}], PlotRange -> All, AspectRatio -> Automatic]