In the present work, we propose new methods for the problem of third-order tensor completion and tensor robust principal component analysis (TRPCA). The proposed approaches are based on finding a low-rank tensor by solving some optimization problems of tensor nuclear norm under some constraints and by using the discrete cosine transform. For the problem of completion, we add some regularization techniques by using the first order and a second-order total variation to enhance the results. Both the main optimization problems lead to some tensor problems that will be solved by using the Alternative Direction Method of Multipliers (ADMM), and also we use for the problem of TRPCA the Proximal Gradient Algorithm to solve it and we will compare the results given by the two ways. We also present some numerical experiments of the proposed methods.