2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210056We show that if a real trigonometric polynomial has few real roots, then the trigonometric polynomial obtained by writing the coefficients in reverse order must have many real roots. This is used to show that a class of random trigonometric polynomials has, on average, many real roots. In the case that the coefficients of a real trigonometric polynomial are independently and identically distributed, but with no other assumptions on the distribution, the expected fraction of real zeros is at least one-half. This result is best possible.5 pages. To appear in PAMSProbabilityComplex Variables60G99; 42A05; 30C15text