I am propagating uncertainties for a lab report in Physics. Anyways, I have a function: $$cos(θ_0 )= \frac{0.725\space m-h}{0.675\space m} $$ The uncertainty of both the numerator and denominator is $\pm 0.1\space$mm. There are multiple values of $h$ but for the sake of demonstration, assume $h=0.05$ m.
How would I calculate the uncertainty of the angle, $\theta_0$? Thanks in advance.
Assuming $h = 0.5 \, \mathrm{m}$,
$$\cos(\theta_0) = \frac{0.725 - 0.5}{0.675} = \frac{1}{3}$$
Using fractional uncertainties (as a consequence of taking derivatives),
\begin{align} \frac{\Delta\cos(\theta_0)}{\cos(\theta_0)} &\approx \frac{\Delta\text{numerator}}{\text{numerator}} + \frac{\Delta\text{denominator}}{\text{denominator}} \\ \implies\Delta\cos(\theta_0) &\approx \frac{1}{3} \left(\frac{0.0001}{0.725 - 0.5} + \frac{0.0001}{0.675}\right)\\ &\approx 0.0002 \end{align}
We can now write \begin{align} \Delta\cos(\theta_0) &= \frac{\cos(\theta_0 + \Delta\theta_0) - \cos(\theta_0 - \Delta\theta_0)}{2} \approx 0.0002\\ \cos(\theta_0 + \Delta\theta_0) - \cos(\theta_0 - \Delta\theta_0) &\approx0.0004 \end{align}
Using compound angle identities, we get $$- 2\sin(\theta_0)\sin(\Delta\theta_0) \approx 0.0004$$
Substituting $\theta_0 = \cos^{-1}(1/3)$, \begin{align} \frac{-4\sqrt{2}}{3}\sin(\Delta\theta_0) &\approx 0.0004\\ \implies \sin(\Delta\theta_0) &\approx \frac{-0.0004\times3}{4\sqrt{2}}\\ \implies |\Delta\theta_0| &\approx \boxed{0.01^{\circ}} \end{align}