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A332500 Decimal expansion of the maximal normal distance between sine and cosine; see Comments 13
1, 0, 9, 4, 9, 9, 8, 9, 8, 4, 3, 7, 0, 8, 7, 2, 4, 2, 8, 6, 5, 0, 4, 0, 8, 3, 0, 0, 7, 1, 5, 5, 2, 4, 6, 7, 1, 2, 9, 1, 0, 5, 1, 4, 0, 6, 0, 7, 0, 5, 4, 3, 6, 0, 2, 0, 6, 5, 8, 0, 3, 3, 4, 2, 9, 5, 5, 1, 8, 7, 5, 4, 4, 9, 6, 2, 2, 1, 4, 0, 5, 4, 1, 3, 0, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Let S and C denote the graphs of y = sin x and y = cos x.  For each point (u, sin u) on S, let S(u) be the line normal to S at (u, sin u), and let (snc u, cos(snc u) be the point of intersection of S(u) and C.  Let d(u) be the distance from (u,sin u) to (snc u, cos(snc u)). We call d(u) the u-normal distance from S to C and note that in [0,Pi], there is a unique number u' such that d(u') > d(u) for all real numbers u except those of the form u' + k*Pi.  We call d(u') the maximal normal distance between sine and cosine, and we call snc the sine-normal-to-cosine function.

For each (v, cos v) on C, let C(u) be the line normal to C at (v, cos v), and let (cns v, sin(cns v)) be the point of intersection of C(u) and S.  Let e(v) be the distance from (v, cos v) to (cns v, sin(cns v)). We call d(v) the v-normal distance from C to S and note that there exists a unique number v' that maximizes e, and e(v') = d(u').  We call cns the cosine-normal-to-sine function. The numbers u' and v' are given in A332501 and A332503.

Note that the maximal normal distance (see Example) exceeds the normal distance from (Pi/2,1) in sine to (Pi/2,0) in cosine - possibly a surprise!

LINKS

Table of n, a(n) for n=1..86.

FORMULA

d(u') = 2*sqrt((u - 3 Pi/4)^2 + (sin u)^2).

EXAMPLE

2.72573705679992524967463858129656... = the number u in [0,2 Pi] such that the line normal to S at (u, sin u) passes through the point (3 Pi/4,0); cf. A332501.

0.4039727532995172093189617400663... = sin u; cf. A086751.

1.0949989843708724286504083007155... = maximal normal distance between sine and cosine.

1.9866519235847646080193264936226... =  snc u; cf A332503.

MATHEMATICA

Plot[{Sin[x], Cos[x]}, {x, -Pi, 3 Pi}, AspectRatio -> Automatic,

ImageSize -> 600, PlotLabel -> "sine and cosine"]

t = Table[x = x /. FindRoot[Cos[x] == -x Sec[u] + u Sec[u] + Sin[u], {x, 0}], {u, -2 Pi, 2 Pi, Pi/101}];

ListPlot[t, PlotLabel -> "y \[Equal] snc(x)"]

ListPlot[Cos[t], PlotLabel -> "y \[Equal] cos(snc (x))"]

t = Table[x = x /. FindRoot[Sin[x] == x Csc[u] - u Csc[u] + Cos[u], {x, 0.1}], {u, -2 Pi + .01, 2 Pi - .01, Pi/101}];

ListPlot[t, PlotLabel -> "y \[Equal] cns(x)"]

ListPlot[Sin[t], PlotLabel -> "y \[Equal] sin(cns(x)"]

u = u /. FindRoot[0 == (-3 Pi/4) Sec[u] + u Sec[u] + Sin[u], {u, 1}, WorkingPrecision ->120]  (* A332501 *)

y = Sin[u]  (* A086751 *)

d = 2*Sqrt[(u - 3 Pi/4)^2 + y^2]  (* A332500 *)

RealDigits[u][[1]]  (* A332501 *)

RealDigits[y][[1]]  (* A086751 *)

RealDigits[d][[1]]  (* A332500 *)

CROSSREFS

Cf. A086751, A332501, A332502, A332503, A332504, A332505, A332506, A332507.

Sequence in context: A258414 A237185 A154201 * A253203 A338151 A255642

Adjacent sequences:  A332497 A332498 A332499 * A332501 A332502 A332503

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, May 05 2020

STATUS

approved

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Last modified April 16 10:45 EDT 2021. Contains 343037 sequences. (Running on oeis4.)