Abstract & Overview
This article explores the use of integral methods in the proof of trigonometric inequalities. Although classical methods such as induction, convexity, and series expansions are well established, the manipulation of integrals to establish sharp trigonometric inequalities remains clearly underexploited. In particular, we use primitive techniques, the Chebyshev integral inequality, and the Jensen integral inequality to recover known results, including the Jordan, Kober, and Cusa-Huygens inequalities. New inequalities are also derived and discussed. The corresponding proofs are given in detail for the sake of completeness. Some figures illustrate selected two-sided inequalities. In final, this article provides a collection of trigonometric inequalities suitable for advanced teaching and further research in many areas of analysis.