A Simple Trigonometric Classification of Quintic Roots

This article provides a simple trigonometric method for determining how many roots of a quintic equation are real and how many are complex, without solving the equation. The approach transforms a depressed quintic $t^5+m{t}^3+n{t}^2+pt+q=0$ with $m<0$ into the trigonometric equation $f(\theta )=\alpha {\cos }^2\text{\hspace{0.17em}⁣}\theta +\beta \cos \theta +\cos 5\theta +\gamma =0$ via the Chebyshev identity $16{\cos }^5\text{\hspace{0.17em}⁣}\theta -20{\cos }^3\text{\hspace{0.17em}⁣}\theta +5\cos \theta =\cos 5\theta$. The derivation is computationally light and conceptually natural, extending the quartic case to fifth-degree equations. As the Abel–Ruffini theorem forbids a general algebraic solution for the quintic, having a simple trigonometric criterion for the nature of its roots is especially appealing.