In the past ten years, trigonometric-type distributions have drawn a lot of attention due to their excellent capacity to assess a wide range of data types thanks to their oscillatory characteristics. In this article, we contribute to the development of such distributions by providing a flexible version of the sine-generated family of distributions. It is elaborated by combining the famous ‘‘sine-generated family’’ with a newly introduced family: the modified power generalized family. In the first part, we study its main theoretical properties, providing the corresponding probabilistic functions, quantiles, and tractable series expansions for moments, among others. An emphasis is put on the member of the family defined with the exponential distribution as the baseline distribution; a new two-parameter lifetime exponential distribution is thus created. The second part is devoted to inference and statistical applications. The maximum likelihood, least squares, weighted least squares, Cramér–von Mises, and Anderson–Darling methods are examined to estimate the unknown parameters of the new exponential distribution. To assess the performance of these estimates, a Monte Carlo simulation is run. Two real-life data sets are analyzed to demonstrate the flexibility of the proposed family. It is found that the new exponential model is more flexible than the comparable models, including the modified power exponential models. In the last part, we use some members of this new family to compute the ruin probabilities in the insurance company problems.