In this paper, we propose a new generalization of the Poisson distribution by using the concept of the weighted distribution; a trigonometric weight with the cosine function is used. We derive some distributional properties of the new distribution, such as the cumulative distribution function, moment generating function, factorial moments, and index of dispersion. Then, the related model is considered for modeling purposes, with estimation of the model parameters performed via several methods. Zero-inflated count regression analysis is introduced by using the new distribution. Finally, we provide two applications of the obtained results on practical data sets.