In this study, we introduce general families of cosine and sine Appell polynomials and investigate their quasi-monomiality properties, recurrence relations, lowering and raising operators, integro-partial lowering and raising operators, differential equations, integro-partial differential equations and determinant forms, summation formulas. These new definitions include some potentially useful subfamilies of polynomials such as general cosine and sine unified Apostol-type polynomials, cosine and sine Gould-Hopper, Laguerre and truncated exponential based unified Apostol-type polynomials. The particular cases of the main results are also given for these subfamilies.