A197003 Decimal expansion of the slope of the line y=mx which meets the curve y=cos(x+Pi/4) orthogonally over the interval [0, 2*Pi] (as in A197002).

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

Offset

1,3

Comments

See the Mathematica program for a graph.

xo=0.3695425666075803208276560438369...

yo=0.4039727532995172093189617400663...

m=1.09316974498501692208815321416057...

|OP|=0.54749949218543621432520415035...

Links

Table of n, a(n) for n=1..101.

Formula

Equals sqrt(2-2*sqrt(1-d^2))/d where d = A003957. - Gleb Koloskov, Jun 16 2021

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]

Prog

(PARI) my(d=solve(x=0, 1, cos(x)-x)); sqrt(2-2*sqrt(1-d^2))/d \\ Gleb Koloskov, Jun 16 2021

Crossrefs

Cf. A003957, A197002, A196996, A197000.

Sequence in context: A187832 A085579 A081813 * A048799 A309893 A188887

Adjacent sequences: A197000 A197001 A197002 * A197004 A197005 A197006

Keyword

nonn,cons

Author

Clark Kimberling, Oct 09 2011

Status

approved