Permutation polynomials are of particular significance in several areas of applied mathematics, such as Coding theory and Cryptography. Many recent constructions are based on the Akbary-Ghioca-Wang (AGW) criterion. Along this line of research, we provide new classes of permutation trinomials in $\mathbb{F}_{q^2}$ of the form $f(x)=x^r h(x^{q-1})$, by studying permutations of the set of $(q+1)$-th roots of unity, which look like monomials on the sets of suitable partitions.