In this article, we propose a new one-dimensional discrete chaotic map, obtained by combining a polynomial logistic map and a sinusoidal map. Dynamic analysis of the proposed map shows that it has better chaotic properties, good ergodicity over a wide range of parameters, and a relatively large key space. Compared to classical logistic and sinusoidal maps, the proposed map exhibits improved ergodicity, with state variables uniformly distributed in the interval [0,1], confirming the dynamic superiority of the proposed map and its suitability for cryptographic and pseudo-random generation applications. Based on these properties, we propose a new image encryption algorithm using sequences from the new chaotic map. The scheme is based on a permutation phase and two a diffusion phase driven by the chaotic sequences generated by the new discrete map. The performance of the proposed system is evaluated through sensitivity tests to initial conditions and keys, key space analysis, and differential attacks. In addition, security indicators such as information entropy, NPCR, UACI, correlation coefficients, and execution time are calculated to validate the effectiveness and robustness of the encryption algorithm.