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 $8{\cos }^4\text{\hspace{0.17em}⁣}\theta -8{\cos }^2\text{\hspace{0.17em}⁣}\theta +1=\cos 4\theta$. 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.