A Simple Trigonometric Classification of Quartic Roots

TL;DR

This paper introduces a straightforward trigonometric approach to classify the roots of a quartic equation as real or complex without solving it, simplifying the traditional discriminant method.

Contribution

It presents an elementary, computationally light trigonometric method that replaces the classical discriminant for root classification of quartic equations.

Findings

Provides a simple trigonometric function analysis for root classification.

Avoids solving the quartic equation directly.

Potentially demystifies the geometric interpretation of quartic roots.

Abstract

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.