In the most recent era, the extensions of the probability models via trigonometry methods have received great attention. This paper also offers a novel trigonometric version of the Weibull model called a type-I cosine exponentiated Weibull (for short “TICE-Weibull”) distribution. The identifiability properties for all three parameters of the TICE-Weibull model are derived. The estimators of the TICE-Weibull model are derived by implementing the maximum likelihood approach. To demonstrate the effectiveness of the TICE-Weibull model, two applications from real-world phenomena are analyzed. In addition, the proposed statistical model is established for an attribute control chart based on a time-truncated life test. The advantage of the developed charts is examined based on the average run length (ARL). The necessary tables of shift sizes and various sample sizes are offered for numerous values of the distribution parameters, as well as specified ARL and shift constants. Some numerical examples are discussed for various scheme parameters to study the performance of the new TICE-Weibull attribute control charts. According to our search and a brief study of the statistical literature, there is no published work on the development of a control chart using new probability models that are introduced using the cosine function. This is the key motivation of this work, which fills this amazing and interesting research gap.