sin(A/2) = sqrt((s-b)(s-c)/(bc)), cos(A/2) = sqrt(s(s-a)/(bc)), tan(A/2) = sqrt((s-b)(s-c)/(s(s-a))). These express half-angle trig ratios entirely in terms of sides (via semi-perimeter). They bridge the gap between side-based and angle-based descriptions of a triangle.