In this paper, based on the separation of variables method from mathematical physics equations, we derive the Fresnel equations from Maxwell’s equations and the boundary conditions at dielectric interfaces. Then using contour plots of the reflection (transmission) coefficient’s modulus and argument vs. relative index of refraction and incident angle, we systematically analyze the conditions for half-wave loss in reflected waves and additional optical path differences in thin-film interference problems. This work underscores the intrinsic link between optics and methods of mathematical physics, serving to stimulate students’ interest in comprehensively applying their knowledge.