The exact pendulum solution uses the Jacobi elliptic function sn(u,k) with modulus k = sin(θ₀/2). Period T = 4√(L/g)·K(k) where K(k) is the complete elliptic integral of the first kind. For small angles, sn(u,k) ≈ sin(u), recovering the harmonic approximation θ ≈ θ₀ cos(ωt).