A332501 Decimal expansion of the number u' in [0,2 Pi] such that the line normal to the graph of y = sin x at (u', sin u') passes through the point (3 Pi/4,0).

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

Offset: 1

Author

Clark Kimberling, May 05 2020

Keywords

• nonn
• cons

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.

The distance from (u',sin u') to its reflection in (3 Pi/4,0) is the maximal normal distance between sine and cosine. This distance is slightly greater than 1. See A332500.

Examples

			2.7257370567999252496746385812...
		

Crossrefs

Cf. A332500, A086751, A332503, A177870, A003957.

Programs

Mathematica

(* See A332500. *)

PARI

3/4*Pi+solve(x=0,1,cos(x)-x)/2 \\ Gleb Koloskov, Jun 17 2021

Formula

Equals (3/4)*Pi + d/2 = A177870 + A003957/2, where d is the Dottie number. - Gleb Koloskov, Jun 17 2021