Uncertainty in measurements: if $x$ has uncertainty $\pm\epsilon$, what is the uncertainty in $\sin x$? I have two questions regarding uncertainties in measurements. 
First, if I have some measured value for $x$ with an uncertainty $\pm e$, what would be the uncertainty in $\sin x$, $\pm\sin e$?  
Then, if I have a range of values from 30 to 46, can I include an uncertainty value of $\pm 2$? Even if it’s not wrong from a mathematical point of view, would it be unsuitable in a physics laboratory report?
 A: First: You don't necessarily get an uncertainty of $\sin(x)\pm\sin(e)$. Remember that the sine function is periodic. So if you have that $x = 0$ and $e = 2\pi$. Then you would not get an uncertainty of $\sin(0) \pm \sin(2\pi) = 0 \pm 0$. In such a case you would get $0\pm 1$ since on the interval $[-2\pi, 2\pi]$ sine takes all values. Now for another example: If you get a value of $x =0.7$ and $e = 0.1$ then indeed you get $\sin(x)\pm \sin(e)$ because $sin$ is strictly increasing on the interval $[0.6, 0.8]$.
Second: I don't think I quite understand, and you might want to ask about some of this on the physics.SE site, but if you are trying to measure something, and you get values ranging from $30$ to $46$, then I guess you might give the average of $38$ with an uncertainty of $\pm 8$. Now, from what I understand, giving uncertainties of measurements in physics is about more than just giving the range of values that you got. You also have to account for uncertainties in the instruments.
Edit: If you have $\sin(x)$ and you measure $x$ in degrees, then $\sin$ is increasing on the interval $[30,46]$. So if the uncertainty is $\pm 2$ on $x$ (from the instrument or what ever), then the uncertainty on $\sin(x)$ is $\pm \sin(2)$.
