In this paper, we obtain the analytical solutions of two kinds of transcendental equations with
numerous applications in college physics by means of the Lagrange inversion theorem. Afterwards we
rewrite them in the form of a ratio of rational polynomials by a second-order Padé approximant from
a practical and instructional perspective. Our method is illustrated in a pedagogical manner for the
benefit of students at the undergraduate level. The approximate formulas introduced in the paper can
be applied to abundant examples in physics textbooks, such as Fraunhofer single-slit diffraction,
Wien’s displacement law, and the Schrödinger equation with single- or double- δ potential. These
formulas, consequently, can reach considerable accuracies according to the numerical results;
therefore, they promise to act as valuable ingredients in the standard teaching curriculum.