Half-angle formulas express trig ratios of A/2, B/2, C/2 entirely in terms of the sides a, b, c through the semi-perimeter s. 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))). An alternative form: tan(A/2) = r/(s-a) = Delta/(s(s-a)). These formulas bridge the gap between side-length data and angle data, enabling conversions in either direction. They are derived by applying the half-angle identities sin^2(A/2) = (1-cosA)/2 and cos^2(A/2) = (1+cosA)/2, then substituting cosA from the cosine rule. The square root is always positive since A/2 is between 0 and pi/2 for any triangle angle.