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

σ

0.7

sin

cos

mean	standard deviation

The mean and standard deviation of

sin(x)

and

cos(x)

, when

x

follows the normal distribution with mean

μ

and standard deviation

σ

, 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

σ

and all the approximations are better for small values of

σ

.