A197002 Decimal expansion of xo, where P=(xo,yo) is the point nearest O=(0,0) in which a line y=mx meets the curve y=cos(x+Pi/4) orthogonally.

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

Offset: 0

Author

Clark Kimberling, Oct 09 2011

Keywords

• nonn
• cons

Comments

See the Mathematica program for a graph.

xo=0.36954256660758032082765604383693...

yo=0.40397275329951720931896174006631...

m=1.093169744985016922088153214160579...

|OP|=0.547499492185436214325204150357...

Links

• Index entries for transcendental numbers.

Crossrefs

Cf. A197003, A196996, A197000, A003957.

Programs

Maple

evalf(solve(cos(x)=x,x)/2, 140);  # Alois P. Heinz, Feb 20 2024

Mathematica

c = Pi/4;
xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A197002 *)
m = 1/Sin[xo + c]
RealDigits[m]  (* A197003 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, 1}], ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All, AspectRatio -> Automatic, AxesOrigin -> Automatic]

PARI

solve(x=0,1,cos(x)-x)/2  \\ Gleb Koloskov, Jun 16 2021

Formula

Equals d/2 = A003957/2, where d is the Dottie number. - Gleb Koloskov, Jun 16 2021