A197006 -id:A197006 - cp's OEIS frontend

A197007 Decimal expansion of the slope of the line y=mx which meets the curve y=cos(x+Pi/6) orthogonally over the interval [0, 2*Pi] (as in A197006).

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

Offset: 1

Clark Kimberling, Oct 10 2011

See the Mathematica program for a graph.
xo=0.460885580860965976987981282513698...
yo=0.553292712300593256734925495541442...
m=1.2004990723879979061250465124427113...
|OP|=0.7201030093885853693640956082816...

Cf. A197006, A197002, A196996, A197000.

c = Pi/6;
xo = x /.  FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A197006 *)
m = 1/Sin[xo + c]
RealDigits[m] (* A197007 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, Pi/2}], ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All,
AspectRatio -> Automatic, AxesOrigin -> Automatic]

A197009 Decimal expansion of the slope of the line y=mx which meets the curve y=cos(x+1) orthogonally over the interval [0, 2*Pi] (as in A197006).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 10 2011

See the Mathematica program for a graph.
xo=0.277097976418521518914833086895...
yo=0.289494183027862650094360757305...
m=1.0447358251025919644670467125044...
|OP|=0.4007370341535820008719293563...

			1.044735825102591964467046712504404113048658932805059578874283182084650....

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A179378, A197002, A196996, A197000.

c = 1;
xo = x /. FindRoot[x == Sin[x + c] Cos[x + c], {x, .8, 1.2}, WorkingPrecision -> 100]
RealDigits[xo] (* A179378 *)
m = 1/Sin[xo + c]
RealDigits[m]  (* A197009 *)
yo = m*xo
d = Sqrt[xo^2 + yo^2]
Show[Plot[{Cos[x + c], yo - (1/m) (x - xo)}, {x, -Pi/4, Pi/2}],
ContourPlot[{y == m*x}, {x, 0, Pi}, {y, 0, 1}], PlotRange -> All,
AspectRatio -> Automatic, AxesOrigin -> Automatic]

default(realprecision, 100); 1/sin(1 + solve(x=0, 2, x-sin(x+1)*cos(x+1))) \\ G. C. Greubel, Nov 16 2018

cp's OEIS Frontend

A197007 Decimal expansion of the slope of the line y=mx which meets the curve y=cos(x+Pi/6) orthogonally over the interval [0, 2*Pi] (as in A197006).

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