This paper introduces a new shear deformation theory, employing the hyperbolic sine function, for exploring the free vibration properties of a novel functionally graded (FG) shell structure. The proposed theory ensures a parabolic distribution of shear strains and stresses across the thickness, with zero values at the top and bottom surfaces, eliminating the requirement for any shear correction factor. This is the first time such an approach has been utilized for studying this type of FG structure. The material properties are assumed to vary gradually across the thickness in the form of a trigonometric function. The proposed FG material stands out due to its excellent rigidity and smooth and continuous variation of the material components through the thickness. This composition has the potential to compensate for the deficiencies found in conventional FG sandwiches. Two types of functionally graded shells are considered: the trigonometric FG-A shell and the trigonometric FG-B shell. The governing equilibrium equations of the FG shell are derived in detail with the principle of virtual work and are solved analytically by the Galerkin method that can cover different boundary conditions. The proposed solution is constrained to rectangular and straight FG plates of uniform cross-section. A wide range of comparative studies is carried out to establish the accuracy and the performance of the present analytical model. A detailed parametric analysis is performed to highlight the influence of the material inhomogeneity parameter, geometry and various boundary conditions on the vibration response. The proposed model has an important role in the design of various vessels and shells