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This paper adopts the extended rational sinh-cosh as well as sine-cosine procedures to find precise solutions to the Hirota equation and Hirota-Maccari equation. It is illustrated that seeking the precise solutions for these equations plays a foremost and effectual role in solving the numerous kinds of PDEs applied in optics, fluid mechanics, plasma physics and solid physics. Furthermore, we are able to obtain some consequences of dark and cusp wave solutions. Besides, two-dimensional and three-dimensional surfaces have been drawn in order to acknowledge the concept of the acquired equations. [ABSTRACT FROM AUTHOR]