The trigonometric reparameterization of modular equations is one of Ramanujan’s methods for simplifying classical modular equations. In this paper, we extend this method from the classical theory ( \(r=2\) ) to the theories of signature \(r=3, 4\) or 6. Ramanujan discovered three trigonometrically parameterized modular equations of degrees 3, 5, and 7 in the classical theory. Here, we present three new trigonometrically parameterized modular equations of degrees 2, 4, and 9 in the classical theory. According to Ramanujan’s “corresponding theories”, seven new trigonometrically parameterized modular equations of degrees 2, 3, 4, 5, 7, 11 and 13 are found in the theory of signature 3; four of degrees 2, 3, 5 and 7 in the theory of signature 4; and four of degrees 2, 3, 5 and 7 in the theory of signature 6. Additionally, this paper provides some examples of cubically solvable modular equations.
