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$\frac{A}{2}$ = sqrt((s-b)$\frac{s-c}{(bc)}$), cos$\frac{A}{2}$ = sqrt$\frac{s(s-a}{(bc)}$), tan$\frac{A}{2}$ = sqrt((s-b)$\frac{s-c}{(s(s-a)}$)). An alternative form: tan$\frac{A}{2}$ = $\frac{r}{s-a}$ = $\frac{Delta}{s(s-a}$). 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$$\frac{A}{2} = $\frac{1-cosA}{2}$ and cos^2$$\frac{A}{2} = $\frac{1+cosA}{2}$, 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.