For transfer operators associated to a Fuchsian group \(\Gamma\), the Fredholm determinant establishes a connection to the Selberg zeta function and, consequently, to the resonances of the Laplace-Beltrami operator associated to \(\Gamma\). The aim of this talk is first to present the basics of Fredholm determinants and then to compute the determinant for a specific class of (transfer) operators. This is then used to approximate the transfer operator of Hecke triangle groups.