If one has an equation such as $$x=-(3.2±0.1)\cos(30.3º±0.2º).$$ How does the error carry to be able to find the value of $x$? I have found that you have to -sine the error in the cosine, but then how do you deal with the value by which the scalar is multiplied, and its error?

up vote 5 down vote accepted

You can use the error propagation formula for the product of two numbers:

$$x=AB$$ $$\Rightarrow\left(\frac{\Delta x}{x}\right)^2=\left(\frac{\Delta A}{A}\right)^2+\left(\frac{\Delta B}{B}\right)^2$$

where $\Delta x$ is the error in $x$ etc.

So in your case, you would identify $$A=3.2\pm0.1$$ and $$B=\cos((30.3\pm0.2)^\text{o})$$ (note that $\Delta B$ is not simply $0.2^\text{o}$, you have to work it out, but it seems you are fine with this).

The more general error propagation formula is (for any function $f(A,B,...)$):

$$\sigma_f^2=\sigma_A^2\left(\frac{\partial f}{\partial A}\right)^2+\sigma_B^2\left(\frac{\partial f}{\partial B}\right)^2+...$$

which may be used to e.g. find the error in the cosine etc. Note though this assumes $A$ and $B$ are independent.

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