A novel proof of the inequality sin x ≤ x

In this paper, a new and insightful proof of the well-known inequality sin x ≤ x , for x ≥ 0 , is presented using some concepts and tools from differential geometry. Unlike standard analytical or calculus-based proofs, this approach draws on geometric intuition and properties of smooth curves, offering a fresh perspective on a classical result. This approach not only provides a rigorous proof, but also enriches the understanding of the inequality by framing it within a broader geometric context, which is accessible to students with a basic knowledge of calculus.