2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/217732We show that high-dimensional analogues of the sine function (more precisely, the d-dimensional polar sine and the d-th root of the d-dimensional hypersine) satisfy a simplex-type inequality in a real pre-Hilbert space H. Adopting the language of Deza and Rosenberg, we say that these d-dimensional sine functions are d-semimetrics. We also establish geometric identities for both the d-dimensional polar sine and the d-dimensional hypersine. We then show that when d=1 the underlying functional equation of the corresponding identity characterizes a generalized sine function. Finally, we show that the d-dimensional polar sine satisfies a relaxed simplex inequality of two controlling terms "with high probability".22 pages and 2 figures, updated to reflect publication in JAT and the DOIClassical Analysis and ODEsMetric Geometry46C05, 52C99, 39B05, 60D05, 42B99text