Analytic solution for the perturbative MLE equation in the near-circular limit

Derive an analytic solution for the maximum-likelihood estimate of the field direction ψ̂ defined in the near-circular perturbative limit by the nonlinear trigonometric equation A cos(ψ̂) − B sin(ψ̂) = C sin(2ψ̂), where A = (1/N) ∑i sin(νi), B = (1/N) ∑i cos(νi), and C = [I2(κ)]/[κ I0(κ) cosh(2μ0)], which arises from the elliptical-cell electrophoretic sensor-redistribution model.

Background

To obtain a tractable expression for the MLE at small eccentricity, the authors expand the elliptic scale factor and derive the estimator condition, which reduces to an equation of the form A cos(ψ̂) − B sin(ψ̂) = C sin(2ψ̂).

They report that they do not have an analytic solution for this equation and instead solve it numerically, motivating the open problem of finding a closed-form solution for ψ̂ under these conditions.