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4 Elementary FunctionsTrigonometric Functions

§4.18 Inequalities

Jordan’s Inequality

4.18.1 2xπsinxx,
0x12π.
4.18.2 xtanx,
0x<12π,
4.18.3 cosxsinxx1,
0xπ,
4.18.4 π<sin(πx)x(1-x)4,
0<x<1.

With z=x+iy,

4.18.5 |sinhy||sinz|coshy,
4.18.6 |sinhy||cosz|coshy,
4.18.7 |cscz|csch|y|,
4.18.8 |cosz|cosh|z|,
4.18.9 |sinz|sinh|z|,
4.18.10 |cosz| <2,
|sinz| 65|z|,
|z|<1.

For more inequalities see Mitrinović (1964, pp. 101–111), Mitrinović (1970, pp. 235–265), and Bullen (1998, pp. 250–254).