# 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?

## 1 Answer

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.