Significant Figures in functions How to extend the idea of significant figures to operations like $\sin\theta$, $\log x$, $\sqrt{x}$, $\cos^{-1} x$, $e^x$ etc. Is there a general rule or method to find/define the no. of significant figures for any given operation?
 A: In the end it always comes down to error estimation. Significant figures are the ones not affected by your estimated error I would say. 
The most basic tool for estimating the error of a function which you feed a value for which you know the error is (Gaussian-) error propagation (small errors). 
$ z = f(x,y) $
$$ \Delta z_i|_{x_i, y_i} = \sqrt{(\frac{\partial f}{\partial x}|_{x_i,y_i} \Delta x_i)^2 + (\frac{\partial f}{\partial y}|_{x_i,y_i} \Delta y_i)^2}, $$ 
where the index $i$ marks a given set of measured values and respective errors marked by $\Delta$.
Example: 
You measured the current $I = (0.55 \pm 0.01)\,\text{A}$ and the voltage $U = (4.32 \pm 0.05) \, \text{V}$. The uncertainties may come from the manual of your measurement equipment. Now you want to know the resistance of the load using $R = f(I,U) = \frac{U}{I} \approx 7.8545$. First you take the derivatives: $\frac{\partial R}{\partial I} = -\frac{U}{I^2}$ and $\frac{\partial R}{\partial U} = \frac{1}{I}$ and plug them into the formula:
$$ \Delta R = \sqrt{\left(-\frac{4.32 \, \text{V}}{(0.55 \, \text{A})^2} 0.01 \, \text{A}\right)^2 + \left(\frac{1}{0.55 \, \text{A}} 0.05 \, \text{V}\right)^2} \approx 0.169 \, \Omega $$
As this kind of error estimation is rather rough, I would always only take one significant number for it and alway round up error estimates. Therefore we take $\Delta R = 0.2 \, \Omega$. As our error is in the first decimal place, everything after the first decimal place is not significant. Therefore we would write the result as $R = (7.9 \pm 0.2) \, \Omega$
If you have not only a value and an error, but data which gives you value and error as mean and standard error, one would just calculate the function on the data and take mean and standard error afterwards.
For a more advanced analysis in the case of not having that data one could for example use a parametric bootstrap. This may be needed if your errors are not "small" compared to the measurement value or if you have a reason to believe that your data is not normal-distributed. 
