The solution of forward or inverse light scattering problems in astrophysical, biological, and atmospheric sensing applications is typically cost prohibitive for real-time applications. For example, given a probability density for the dimensions, refractive index, and wavelength, evaluating the expected scattering involves an integral over such parameters, and the number of scattering problems solved increases dramatically. In the case of dielectric and weakly absorbing spherical particles (both homogeneous and layered), we begin by highlighting a circular law that restricts scattering coefficients to a circle in the complex plane. Later, the Fraunhofer approximation of Riccati-Bessel functions is used to reduce the scattering coefficients into simpler nested trigonometric approximations. This results in relatively small errors of oscillatory signs that cancel out without a loss of accuracy in the integrals over scattering problems. Thus, the cost of evaluating the two spherical scattering coefficients for any mode is reduced by large factors ≈50, with a larger increase in the speed of the overall computation, as the approximations can be reused for multiple modes. We analyze the errors of the proposed approximation and present numerical results for a set of forward problems as a demonstration.