GUM versus Exact Uncertainty of sin(x) and cos(x) When x Is Normally Distributed
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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 .
Contributed by: M. D. Mikhailov and V. Y. Aibe (June 2012)
(Instituto Nacional de Metrologia, Qualidade e Tecnologia (Inmetro), Brazil)
Open content licensed under CC BY-NC-SA
Snapshots
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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