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# GUM versus Exact Uncertainty of sin(x) and cos(x) When x Is Normally Distributed

The mean and standard deviation of and , when follows the normal distribution with mean and standard deviation , are shown on four plots to compare the approximate and exact solutions. The red curve is an approximation calculated by formulas given in GUM (Guide to Uncertainty in Measurement). The blue curve is exact, as calculated by Mathematica 8. The approximate means are independent of and all the approximations are better for small values of .

### DETAILS

The approximate means and standard deviations of and , where is a normal random variable with mean and standard deviation are
,
,
,
.
The exact means and standard deviations as calculated by Mathematica 8 are
,
,
,
.
Reference
[1] "Evaluation of Measurement Data—Guide to the Expression of Uncertainty in Measurement." BIPM. (Sept 2008) www.bipm.org/utils/common/documents/jcgm/JCGM_100_ 2008_E.pdf.

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