Optical intensity distributions in the focal region play an important role in many optical systems. In this paper, the tunable focusing properties of linearly polarized hyperbolic-cosine-Gaussian beams with sine-azimuthal variation wavefront were investigated by adding a spiral optical vortex. It was found that the focal patterns can be altered very considerably by changing the charge number of the optical vortex under a different phase parameter that indicates the phase change frequency upon increasing the azimuthal angle. The symmetry of focal patterns also changes remarkably upon increasing the charge number. And some novel focal patterns may appear, including a multiple-peak array, wheel focal pattern, or swallow-tailed focal pattern.