I am performing numerical and uncertainty analysis for an oblique shockwave angle function:
$$ \tan(\delta)=\frac{2}{\tan(\theta)}\frac{M^2\sin^2(\theta)-1}{M^2(\gamma+\cos(2\theta))+2} $$
where $\delta$ is deflection angle, $\theta$ is shock angle, M is mach number, and $\gamma$ is the ratio of specific heats. In our observation, $\delta$, $\theta$, and $M$ all have uncertainty. $\gamma$ is assumed to be exact.
Our unknown quantity is $\theta \pm \sigma_{\theta}$. If it was possible to solve for $\theta$ analytically, it would be trivial to propagate uncertainty (excluding covariances):
$$ \sigma_{\theta}^2 = \left( \frac{\partial\theta}{\partial M}\sigma_M \right)^2+ \left( \frac{\partial\theta}{\partial\delta}\sigma_{\delta} \right)^2 $$
The question is: how do I define $\theta$ so that I can take a derivative of it with respect to $M$ and $\delta$ for uncertainty analysis? Or maybe there is a way to approximate $\theta$ and take a numerical derivative? I am probably overthinking this.