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Recently trigonometric inequalities of N. Cusa and C. Huygens (see, e.g., [9]), J. Wilker [11], and C. Huygens [4] have been discussed extensively in mathematical literature. We shall demonstrate that Wilker’s inequality, Huygens’ inequality, and some other related inequalities all follow from the Cusa-Huygens inequality. A generalization of the latter result is also obtained. The hyperbolic counterparts of those inequalities are also derived. Mathematics subject classification (2010): Primary: 26D05, 26D07; secondary: 33B10.

On some inequalities involving trigonometric and hyperbolic functions with emphasis on the Cusa-Huygens, Wilker, and Huygens inequalities