A215668 Decimal expansion of the zero z in (0,Pi/2) of the function sin(sin(x))/sin(x) - cos(cos(x))/cos(x).

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

Offset

0,1

Comments

It is proved (see Witula et al.'s reference) that the function h(x) := sin(sin(x))/sin(x) - cos(cos(x))/cos(x) is decreasing in the interval (0,Pi/2) and has zero z in (0,Pi/4). We have sin(sin(z))/sin(z) = cos(cos(z))/cos(z) = 0.933396189408898411846964... . Moreover the following relation hold: F(z) = min{F(x): x \in R} = 0.10712694487..., where F(x) := cos(sin(x)) - sin(cos(x)) - see also A215670 and the Witula et al.'s reference for more information.

Links

Table of n, a(n) for n=0..108.

Roman Wituła, Danuta Jama, Edyta Hetmaniok, and Damian Słota, On some inequality of the trygonometric type, Zeszyty Naukowe Politechniki Slaskiej - Matematyka-Fizyka (Science Fascicle of Silesian Technical University - Math.-Phys.), 92 (2010), 83-92.

Example

z = 0.692728570186833888... rad  (= 39.6904234198376057055880... deg).

Mathematica

RealDigits[x /. FindRoot[Sin[Sin[x]]/Sin[x] - Cos[Cos[x]]/Cos[x], {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Mar 27 2026 *)

Crossrefs

Cf. A215670, A215832, A215833, A168546, A216891.

Sequence in context: A254135 A198676 A198616 * A344230 A010502 A254292

Adjacent sequences: A215665 A215666 A215667 * A215669 A215670 A215671

Keyword

nonn,cons

Author

Roman Witula, Aug 20 2012

Extensions

Name and comment corrected by Amiram Eldar, Mar 27 2026

Status

approved