Motivated by the classical cosine rule from the trigonometric geometry, a novel generalization of the Sombor index is proposed. The Cosine-Rule Generalized Sombor index \( CoRSO_\theta \) is defined via the expression \( \sqrt{d(u)^2 + d(v)^2 - 2d(u)d(v) \cos \theta} \) where \( d(u) \) and \( d(v) \) denote the degrees of adjacent vertices, and \( \cos \theta \) is the cosine modulator of the degree interaction. The recently proposed variable Euler-Sombor topological index \(EU(\lambda, G)\) defined via the expression \( \sqrt{d(u)^2+d(v)^2+\lambda d(u)d(v)} \) with restricted parameter \( \lambda \in [-2,2] \) is derived from the new index. The functional generalization of the Sombor index is proposed. Mathematical properties of \(CoRSO_\theta\) index are established and its chemical applicability is demonstrated.