The manipulator with five degrees of freedom (5-DOF) cannot reach any pose, so it is easy to get no solution when the traditional pose description method is used to solve the inverse kinematics. In this paper, a 5-DOF manipulator consisting of four rotary joints and one prismatic joint, which is suitable for cleaning, spraying, welding and other operations, is taken as an example to establish a kinematic model and its inverse kinematic analysis is carried out. Then, an inverse kinematics solution method based on joint angle parameterization combined with the feasible direction of approach vector is proposed based on the end pose description of degree of freedom constraints. In this method, firstly, the motion space of the end executor is reduced from three dimensions to two dimensions through joint angle parameterization. Then, the feasible direction of the approach vector of the end executor at different target positions (distal, middle and proximal) is analyzed by geometric method, so as to avoid blind parameter setting and ensure the existence of inverse kinematics solutions. Thereafter, the optimal solution is selected according to the motion continuity and motion range of each joint. The simulation analysis results of path planning show that the actual path is very consistent with the planned path and the joint motion is stable, which proves the feasibility and accuracy of the proposed solution method. The 5-DOF manipulator studied in this paper is representative to some extent and the proposed solution method possesses the advantage of low computational complexity and simple solving process. Therefore, the solution idea can provide reference for the inverse kinematics solution of the manipulator with less degrees of freedom.