In this paper we study the relationships among the exponential spectra of the cosets of an element of a Banach algebra in some quotient Banach algebras. These relationships are necessary to be studied since the spectra and exponential spectra do not always coincide. This is anextensionof some work done by Laura Burlando [1] on the relationships among the spectra of the cosets of an element of a Banach algebra in some quotient algebras.