Uncertainty in a cosine of a measured angle (${\sigma }_S^2={\cos }^2(q\theta ){\sigma }_A^2+[qA\sin (q\theta ){]}^2{\sigma }_{\theta }^2$)

For $S=A\cos (q\theta )$ with independent standard uncertainties, linear propagation gives ${\sigma }_S^2={\cos }^2(q\theta ){\sigma }_A^2+[qA\sin (q\theta ){]}^2{\sigma }_{\theta }^2$. Angular uncertainty must be in radians when using this derivative. Covariance adds cross terms for correlated inputs. Strict error bounds instead add absolute first-order contributions; they do not justify Gaussian significance tests.