The propagation characteristics of annular cos-Gaussian beams and Bessel-Gaussian beams in turbulence have been widely investigated. However, the expression of the average intensity of Bessel-Gaussian beams involves a complex double integral, which requires large computational resources and does not easily ensure accuracy. Under this background, through the general cos-Gaussian beam rotation, a new annular cos-Gaussian beam is proposed which not only has some properties of the general cos-Gaussian beam, but also properties similar to those of the Bessel-Gaussian beam. Moreover, the annular cos-Gaussian beam is simpler in form than the general one and much more time-saving in calculation than the Bessel-Gaussian beams. Based on the extended Huygens-Fresnel principle, the average intensity on a receiver plane of the annular cos-Gaussian beams can be simplified to a summation of several polynomials and it is in good agreement with that of low-order Bessel Gaussian beams. The average radius and the power in the bucket are also derived. In summary, the annular cos-Gaussian beam has promising applications for its quick calculation and simple form among annular beams and hollow beams.