A215670 Decimal expansion of the min value of F(x) := cos(sin(x)) - sin(cos(x)), x in R.

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

Offset: 0

Author

Roman Witula, Aug 20 2012

Keywords

• nonn
• cons

Comments

We note that dF(x)/dx = (-1/2)*h(x)*sin(2*x), x in (0,Pi/2), where h(x) is the function discussed in comments to A215668 (see also Witula et al.'s reference for more informations).

Examples

			min{F(x): x in R} = F(z) = 0.1071269448729529961...
		

References

• R. Witula, D. Jama, E. Hetmaniok, D. Slota, On some inequality of the trigonometric type, Zeszyty Naukowe Politechniki Slaskiej - Matematyka-Fizyka (Science Fascicle of Silesian Technical University - Math.-Phys.), 92 (2010), 83-92.

Crossrefs

Cf. A215668, A215832, A215833, A168546, A216891.

Formula

F(z) = cos(sin(z)) - sin(cos(z)) = (cos(z) - sin(z))*(cos(cos(z)) + sin(sin(z)))*cos(cos(z))/(cos(sin(z)) + sin(cos(z)))*cos(z) = cos(2*z)*cos(cos(z))^2/(cos(sin(z)) + sin(cos(z)))*cos(z)^2 = (1 - tan(z)^2)*cos(cos(z))^2/(cos(sin(z)) + sin(cos(z))), where z := A215668.