I am trying to do a measurement uncertainty calculation. I have a gaussian distributed phase angle (theta) with a mean of 0 and standard deviation of 16.6666 micro radians. The variance is the square of the standard. The formula for the measurment uses cos(theta) in the calculation. I need to know the mean, the variance and the distribution function that result from taking the cosine of theta in order to do my calculations correctly. 



Resultant probability distribution when taking the cosine of gaussian distributed variable
I wrote out the first few terms in the power series for $ \cos \theta $
and then the first few terms of the series for $ \cos^2 \theta .$
 I used your hypothesis of normal distribution, the mean of $ \theta $ is $ \mu = 0$ while the
variance is some $ \sigma^2 .$


Then I looked up the expected values of $ \theta^2, \; \theta^4, \; \theta^6, \; \theta^8 $ at
<http://en.wikipedia.org/wiki/Gaussian_distribution#Moments>
and used that to find good approximations for your new mean $\mu\_1$ and variance $\sigma\_1^2$ in
$$ \mu\_1 = E[ \cos \theta ] = 1 - \frac{\sigma^2}{2} + \frac{\sigma^4}{8} - \frac{\sigma^6}{48} + \cdots $$ 
and
$$ \mu\_1^2 + \sigma\_1^2 = E[ \cos^2 \theta ] = 1 - \sigma^2 + \sigma^4 - \frac{2 \sigma^6}{3} + \cdots $$ 
So when you subtract you get $ \sigma\_1^2 \approx \frac{\sigma^4}{2} $


I will think about it some more, there is a large theory for calculating moments. But I do not see much to be done in the way of an explicit pdf or cdf. 


