  In this article we consider a method of proving a class of inequalities of
the form (1). The method is based on the precise approximations of the sine and
cosine functions by Maclaurin polynomials of given order. By using this method
we 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)].
