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In this paper, we propose a useful method without adding any extra parameters to obtain new probability distributions. The proposed family is a combination of the two existing families of distributions and is called a weighted sine-<i>G</i> family. A two-parameter special member of the weighted sine-<i>G</i> family, using the Weibull distribution as a baseline model, is considered and investigated in detail. Some distributional properties of the weighted sine-<i>G</i> family are derived. Different estimation methods are considered to estimate the parameters of the special model of the weighted sine-<i>G</i> family. Furthermore, simulation studies based on these different methods are also provided. Finally, the applicability and usefulness of the weighted sine-<i>G</i> family are demonstrated by analyzing two data sets taken from the engineering sector.