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In this article we consider a method of proving a class of inequalities ofthe form (1). The method is based on the precise approximations of the sine andcosine functions by Maclaurin polynomials of given order. By using this methodwe present new proofs of some inequalities from the articles C.-P. Chen, W.-S.Cheung [J. Inequal. Appl. 2012:72 (2012)] and Z.-J. Sun, L. Zhu [ISRN Math.Anal. (2011)].

A Method of Proving a Class of Inequalities of Mixed Trigonometric Polynomial Functions