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Statistical methods have widespread applications in real-life scenarios. Researchers have expressed keen interest in implementing different functions to raise new probability distributions. However, very limited efforts have been made to raise new methodologies by applying trigonometric functions. In this paper, we propose a new method based on the trigonometric function. The suggested trigonometric-based method is called a new trigonometric cosine-G (NTC-G) family to generate flexible models. The new method is developed using the cosine function. A subcase of the NTC-G method termed as a new trigonometric cosine-Weibull (NTC-Weibull) is studied. Certain mathematical properties of the NTC-Weibull model are presented. For the NTC-Weibull model, the point estimators are acquired. Finally, the suitability of the NTC-Weibull distribution is demonstrated in light of a healthcare data set. Using the numerical and graphical approaches, the evaluation results of the NTC-Weibull distribution are compared with the Weibull and its four well-known existing modifications.