A Simple Trigonometric Classification of Quartic Roots

This article provides a simple trigonometric method for determining how many roots of a quartic equation are real and how many are complex, without solving the equation. The approach replaces the quartic's classical discriminant -- a degree-six polynomial in the coefficients -- with an elementary analysis of the function $f(\theta) = a\cos\theta + \cos 4\theta + b$ on $[0,\pi]$, obtained by matching the quartic to the Chebyshev identity $8\cos^4\!\theta - 8\cos^2\!\theta + 1 = \cos 4\theta$. The derivation is computationally light and conceptually natural, and has the potential to demystify the geometry of a quartic equation's roots from a trigonometric perspective.