Let's say I have the following equation in which the unknown is $θ$: $$tan(θ)=\frac{a}{b}$$ $$tan(θ)=\frac{5}{3}$$ $$θ=arctan(1.667)$$ $$θ=59.036°$$
Let's say the absolute errors ($∆S$) and percentage errors ($δ$) are the following: $$∆a =0.2 $$ $$∆b =0.3 $$
$$δ_a =4 \text% $$ $$δ_b =10 \text% $$
I want to determine the percentage error ($δ$) in calculating $θ$, how do I do this?
I understand that for products and quotients, the percentage error is summed ($δ_a + δ_b$) and the percentage error for trig functions is of the format $δ =\frac{tan(θ+∆θ)-tan(θ)}{tan(θ)}*100$ but I am not sure how to proceed from here since $θ$ was calculated and not a known value with known absolute or relative errors and I am interested in the percentage error of $θ$ and not $tan(θ)$.
Someone please help, I'm melting in uncertainty abyss here :(
$\endgroup$1 Answer
$\begingroup$We have $\frac {\partial}{\partial x}\arctan (\frac xy)=\frac y{x^2+y^2}$ so the error in the angle due to error in $a$ is $\frac b{a^2+b^2}\Delta a$ You can do a similar thing for errors in $b$ and add them together to get the total error in $\theta$
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