Despite the availability of continuous distributions, few utilize the modeling potential of trigonometric functions, and none are based on the hyperbolic secant function. To fill this gap, we introduce the hyperbolic Sec-B family of trigonometric distributions. As a specific application, we introduce an adapted half-power logistic distribution (HS-PHLD) that retains a simple two-parameter form while offering greater versatility, particularly in tail behavior and skewness. Our research comprehensively explores this, establishing its fundamental mathematical properties, providing series expansions for its functions, and using both non-Bayesian and Bayesian estimation techniques. Monte Carlo simulations are used to validate the effectiveness of these estimators. Practically speaking, the HS-PHLD outperforms well-established models on three real-world datasets from the engineering and survival domains.