I am doing a physics project, and am using the sine law to find the angle of my final momentum vector. Using absolute error arithmetic (i.e. add absolute error when adding/subtracting, add relative error when multiplying/dividing), I have arrived at the expression
$$\sin \alpha = 0.503 \pm 0.08.$$
How do I find the absolute error $x$ for my final value of $\alpha = 30^\circ \pm x$? If there is a rule for this (which I cannot seem to find on the internet), does it hold for logarithms, normal trig functions, etc.?
Best Answer
We have $\alpha=\arcsin{(0.503 \pm 0.08)}$ from your original statement. As $\arcsin(x)$ is an increasing function we can then write $\arcsin{(0.503 - 0.08)} \le \alpha \le \arcsin{(0.503 + 0.08)}$ or approximately, $25.0^\circ \le \alpha \le 35.7^\circ$. We can calculate the uncertainty of $\alpha$ = $\frac{1}{2} \times$ range = $\frac{1}{2} \times (35.7-25.0)=5.32^\circ$. So approximately we have: $$\alpha=30.3^\circ \pm 5.32^\circ$$
In general for an increasing function $f(x)$, if we have $f^{-1}(x)=a\pm b$, then $$x=\frac{f(a+b)+f(a-b)}{2} \pm \frac{|f(a+b)-f(a-b)|}{2}$$