This study introduces the cotangent- $\mathrm{\Phi }$ (NC- $\mathrm{\Phi }$) distribution, which establishes a new type of probability distribution for statistical modeling complex real-world data. The proposed method uses trigonometric transformations to systematically manipulate the skewness and tail characteristics of traditional distributions, thus overcoming the difficulties that often arise in existing models. A complete mathematical theory is formulated, with series expansions of the cumulative distribution function and the probability density function. Based on the Weibull distribution, a new cotangent-Weibull distribution (NCWD) is introduced, and its major statistical features are investigated. A machine learning layer was developed to complement the analytical properties of the formulated NCWD. Parameter estimation is carried out in both frequentist and Bayesian approaches, and a Monte Carlo simulation study is conducted to evaluate and compare the performance of the estimators under different sample sizes. The validity of the proposed model is established through applications to three different real-world datasets: the active repair times (in hours) of an aerial communication transceiver, the inter-arrival times between vehicles passing a fixed point on a roadway, and the susceptibility index data of peppermint packages irradiated by gamma and microwave radiation under non-choice packaging conditions. In each scenario, the NCWD outperforms several other competing models in terms of the goodness-of-fit, thus establishing its appropriateness for reliability, transportation, and irradiation data analyses.