A new simple hyperbolic-type three-dimensional autonomous chaotic system is proposed. It is of interest that the chaotic system has only five terms which mainly rely on a nonlinear quadratic hyperbolic sine term and a quadratic cross-product term. Compared with other three-dimensional chaotic systems, the new system has not only less terms, but also a wider range of chaos when the parameter varies. Basic dynamical properties of the system are studied by numerical and theoretical analysis. Moreover the projective synchronization of the five-term hyperbolic-type chaotic system with fully uncertain parameters is also investigated in this paper. Based on Lyapunov stability theory and Barbalat's lemma, a new adaptive controller with parameter update law is designed to projectivly synchronize two chaotic systems asymptotically and globally, including two identical exponential-type chaotic systems and two non-identical chaotic systems. Numerical simulations show the effectiveness and the feasibility of the developed methods.