# Propagation of uncertainties while calculating trigonometric ratios

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?

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.