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.

$\begingroup$ Your angles are small enough that sin(theta) is very close to theta, so you can simplify your analysis by approximating cos(theta) with sqrt(1theta^2). $\endgroup$ – Tracy Hall Aug 11 '10 at 20:12

3$\begingroup$ The other reasonable approximation is $$ \cos \theta \approx 1  \frac{\theta^2}{2} $$ which will definitely give you a mean in closed form. $\endgroup$ – Will Jagy Aug 12 '10 at 3:18
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.
A quick way to find the mean of $\cos(\theta)$, where $\theta\sim \mathcal{N}(0, \sigma^2)$, is through calculating the mean of a complex variable $e^{j\theta}=\cos(\theta)+j\sin(\theta)$. We have
$E [e^{j\theta}]=e^{0+(j\sigma)^2/2}=e^{\sigma^2/2}$
which implies that the mean of the imaginary part $E [\sin(\theta)]$ equals zero and the mean of the real part $E[\cos(\theta)]$ equals $e^{\sigma^2/2}$.
The answer $\mu_1$ derived by Will Jagy is in fact the Taylor series expansion of $e^{\sigma^2/2}$.
The variance of $\cos(\theta)$ can be obtained as:
$E[\cos^2(\theta)]E[\cos(\theta)]^2= E[\frac{1}{2}+\frac{\cos(2\theta)}{2}] E[\cos(\theta)]^2= \frac{1}{2}[1e^{\sigma^2}]^2$

1$\begingroup$ If anyone else is looking for this: I used this to work out the mean and variance of $\cos x$ and $\sin x$ where $x \sim \mathcal{N}(\mu, \sigma^2)$ at this link. Nothing complicated, just a bunch of trig identities that I haven't remembered since tenth grade. $\endgroup$ – Dougal Jan 20 '14 at 5:19

2$\begingroup$ For completeness, adding a reference to the calculation of mean of a complex variable when the exponent terms follow the normal distribution: math.stackexchange.com/a/2050140/29735 $\endgroup$ – AruniRC Dec 29 '17 at 18:25
Hi, I know this was asked a long time ago but I have just discovered it because I require a similar solution. It is possible to generate an expression, albeit as an infinite summation. For practical purposes, the first few terms of the summation should suffice.
Let $X$ denote a random variable with pdf $f_X(x)$. Let $Y=g(X)$ be a function of $X$. We can specify the cdf of $Y$, denoted $F_Y(y)$ as follows:
$F_Y(y)=\mathbb{P}(g(X)\leq y)=\int\limits_{\Omega}f_X(x)\text{d}x$,
where the domain of integration $\Omega$ is defined as
$\Omega=\left\lbrace x:g(x)\leq y \right\rbrace$
In our case, $g(x)=\cos x$, so we need an expression for the domain of $x\in\mathbb{R}$ such that $\cos x\leq y$. This is given by
$2k\pi+\arccos(y) \leq x < 2(k+1)\pi\arccos(y)\, k\in\mathbb{Z}$
So integrating over this domain, we obtain
$F_Y(y)=\sum\limits_{k=\infty}^{\infty} \int\limits_{2k\pi+\arccos(y)}^{2(k+1)\pi\arccos(y)} f_X(x)\text{d}x$
Now in our case $X\sim\mathcal{N}(0,\sigma)$, so
$f_X(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\exp\left(\dfrac{x^2}{2\sigma^2}\right)$
and the integral of this pdf between limits is given by the cdf of the normal distribution, which we denote $\Phi$:
$\int\limits_{a}^{b}f_X(x)\text{d}x = \Phi(b/\sigma)\Phi(a/\sigma)$
The cdf of $Y$ is therefore
$F_Y(y)=\sum\limits_{k=\infty}^{\infty} \Phi\left(\dfrac{2(k+1)\pi\arccos(y)}{\sigma}\right)  \Phi\left(\dfrac{2k\pi\arccos(y)}{\sigma}\right)$
To compute the pdf, take the derivative with respect to $y$:
$f_Y(y)=\dfrac{dF_Y(y)}{dy} = \sum\limits_{k=\infty}^{\infty} \dfrac{1}{\sqrt{1y^2}}\left( f_{X}(2(k+1)\pi\arccos(y) ) + f_{X}(2k\pi+\arccos(y)) \right)$
There are probably better ways to do this. It's possible the final summation can be rewritten or simplified. But this seems to match with a numerical check.

$\begingroup$ Think it should be $$F_Y(y) = \sum_{k=\infty}^\infty \Phi\left( \frac{2(k+1) \pi  \arccos(y)}{\sigma} \right)  \Phi\left( \frac{2k\pi + \arccos(y)}{\sigma} \right)$$, no? (Note the addition instead of subtraction in the second $\Phi$ argument.) This makes the pdf $$f_Y(y)=\dfrac{dF_Y(y)}{dy} = \sum\limits_{k=\infty}^{\infty} \dfrac{1}{\sqrt{1y^2}}\left( f_{X}(2(k+1)\pi\arccos(y) )  f_{X}(2k\pi+\arccos(y)) \right)$$. $\endgroup$ – Dougal Jan 24 '14 at 22:47
Original Approach
Given a normal distribution with mean $\mu$ and variance $\sigma^2$, $X = \mathcal{N}(\mu,\sigma^2)$, if you pass it through trigonometric functions, you can approximate the result with the new normal distributions below
1) normal distribution passed through Cosine function:
$X_{\cos} = \mathcal{N}(\cos(\mu),\sigma^2\sin^2(\mu))$
so the new average is $\cos(\mu)$ and the new standard deviation is $\sigma\sin(\mu)$.
2) normal distribution passed through a Sine function:
$X_{\sin} = \mathcal{N}(\sin(\mu),\sigma^2\cos^2(\mu))$
so the new average is $\sin(\mu)$ and the new standard deviation is $\sigma\cos(\mu)$.
The Matlab script that I used to find these relations is below.
%% Cody Martin
% 9/2/2010
% mfile used to discover the mean and variance of a normal distribution
% passed through cosine and sine functions...results:
%  N(mu,sig^2) > cos(N(mu,sig^2)) = N(cos(mu),sig^2*sin^2(mu))
%  N(mu,sig^2) > sin(N(mu,sig^2)) = N(sin(mu),sig^2*cos^2(mu))
%% distribution of cosine and sine of a normal distribution?
cresults = zeros(0,5);
sresults = zeros(0,5);
% loop from an average angle 90 degrees to +90 degrees
for theta = pi/2:pi/180:pi/2
theta1sig = pi/36; % standard deviation of orinigal normal distribution
vtheta = theta + theta1sig*randn(99999,1); % create 99999 points using this avg and std
vctheta = cos(vtheta); % take the cosine of those points
vstheta = sin(vtheta); % take the sine of those points
theta_ = min(vtheta):0.01:max(vtheta); % for plotting ideal distributions
ctheta_ = min(vctheta):0.01:max(vctheta); % for plotting
stheta_ = min(vstheta):0.01:max(vstheta); % for plotting
figure(1); clf;
subplot(211); hold on;
plot(theta_,cdf('normal',theta_,theta,theta1sig),':'); % plot cdf of normal distribution with avg and std
plot(sort(vtheta),[1:length(vtheta)]/length(vtheta)); % plot cdf of 99999 points
plot(sort(vctheta),[1:length(vctheta)]/length(vctheta),'k','LineWidth',2); % plot cdf of cos(99999 points)
plot(ctheta_,cdf('normal',ctheta_,cos(theta),... % plot cdf of norm dist with new avg and std after being passed through cos()
sqrt(theta1sig^2*sin(theta)^2)),'r:');
plot(cos(theta)*[1 1],[0 1],'k:'); % vertical line @ cos(theta)  shows new average matches cos(old avg)
title('Cosine of a Normal Distribution (for Different Initial Averages)');
legend('Norm CDF Theory','Norm CDF 99999','Cos(Norm CDF 99999)','Cos(Norm CDF) Theory');
axis([pi/2 pi/2 0 1])
subplot(212); hold on;
plot(theta_,cdf('normal',theta_,theta,theta1sig),':');
plot(sort(vtheta),[1:length(vtheta)]/length(vtheta));
plot(sort(vstheta),[1:length(vstheta)]/length(vstheta),'k','LineWidth',2);
plot(stheta_,cdf('normal',stheta_,sin(theta),...
sqrt(theta1sig^2*cos(theta)^2)),'r:');
plot(sin(theta)*[1 1],[0 1],'k:');
title('Sine of a Normal Distribution (for Different Initial Averages)');
legend('Norm CDF Theory','Norm CDF 99999','Sin(Norm CDF 99999)','Sin(Norm CDF) Theory');
axis([pi/2 pi/2 0 1])
% fprintf('theta: %3.0f\tstd: %5.3f\tsin(theta): %5.3f\tavg: %5.3f\tstd: %5.3f\n',theta*180/pi,theta1sig,sin(theta),mean(vstheta),std(vstheta));
cresults = [cresults; theta theta1sig cos(theta) mean(vctheta) std(vctheta)];
sresults = [sresults; theta theta1sig sin(theta) mean(vstheta) std(vstheta)];
end
figure(2); clf;
subplot(211); hold on;
plot(cresults(:,1),cresults(:,end));
plot(cresults(:,1),abs(theta1sig*sresults(:,3)),'r:');
title('Standard Deviation of Cosine of a Normal Distribution as a Function of the Original Average');
legend('From 99999 Points','Fit: std = \sigmasin(\mu)');
ylabel('std(cos(\theta_{vector})) [rad]');
xlabel('\theta [rad]');
subplot(212); hold on;
plot(sresults(:,1),sresults(:,end));
plot(sresults(:,1),abs(theta1sig*cresults(:,3)),'r:');
title('Standard Deviation of Sine of a Normal Distribution as a Function of the Original Average');
legend('From 99999 Points','Fit: std = \sigmacos(\mu)');
ylabel('std(sin(\theta_{vector})) [rad]');
xlabel('\theta [rad]');
figure(3); clf;
subplot(211); hold on;
plot(cresults(:,1),abs(theta1sig*sresults(:,3))cresults(:,end));
title('Error Between \sigma^2sin^2(\mu) and std of 99999 Draws of cos(\theta)')
ylabel('Residual [rad]');
xlabel('\theta [rad]');
subplot(212); hold on;
plot(sresults(:,1),abs(theta1sig*cresults(:,3))sresults(:,end));
title('Error Between \sigma^2cos^2(\mu) and std of 99999 Draws of cos(\theta)')
ylabel('Residual [rad]');
xlabel('\theta [rad]');
Update
As others have pointed out, this fails where $\cos(\mu)$ and $\sin(\mu)$ are near 0. Residuals between my proposed solution and the empirical results from 99999 draws are shown below.

$\begingroup$ Cody, is it wright what you say? Sigma is varying with the mean? If I measure an angle of 90 degrees, then $N_{\cos}(0,{\sigma}^2)$ and $N_{\sin}(1,0)$? And if I measure an angle of 0 degrees, then $N_{\cos}(1,0)$ and $N_{\sin}(0,{\sigma}^2)$ ? Where do I find the theory of that? $\endgroup$ – user24033 May 28 '12 at 14:21

$\begingroup$ Cody, I'm afraid your answer is incomplete. The problem I see lays with the variance. If $X \sim N(\mu,\sigma^2)$ indeed results with $cos(X) \sim N(cos(\mu), \sigma^2 sin^2(\mu))$ then for, e.g., $\mu = \frac{\pi}{2}$ the approximation is $N(1,0)$ regardless of $\sigma$. This seems to be a poor approximation because an increase in the variance of $X$ should always result in an increase of the variance of $cos (X)$. $\endgroup$ – user33615 May 2 '13 at 5:04
Stochastic Calculus Approach:
If $W_t$ is a standard Wiener process, we know that the increment $W_t=W_0$ is normally distributed (with mean 0 and variance $t$). Let $ \begin{equation} f(t,x):=e^{t/2}cos(x) \end{equation} $ Then by Ito's lemma we have that $X_t:=f(t,W_t)$ satisfies: \begin{equation} e^{t/2}cos(W_t) = 1+ \int_0^t \frac{e^{t/2}}{2}sin(W_t)dW_t. \end{equation}
 Mean Taking expectation on both sides of the above equation yields: \begin{align} \mathbb{E}[e^{t/2}cos(W_t)] & = \mathbb{E}[1+ \int_0^t \frac{e^{t/2}}{2}sin(W_t)dW_t] \end{align} since $\int_0^t sin(W_t)dW_t$ is a stochastic integral, it must be a martingale; wherefrom it follows that the above expectation reduces to: \begin{align} \mathbb{E}[e^{t/2}cos(W_t)] & = \mathbb{E}[1]+ \mathbb{E}[\int_0^t \frac{e^{t/2}}{2}sin(W_t)dW_t] = 1 \\ \therefore \mathbb{E}[cos(W_t)] & = e^{t/2}. \end{align} Since $\theta:=W_tW_0$ is a $\mathscr{N}_1(0,t)$distributed random variable. Then we may conclude that: \begin{align} \mathbb{E}[cos(\theta)] & = e^{t/2}. \end{align}
 Variance: To calculate the variance we note that by the Ito isometry we have that: \begin{align} \mathbb{E}[\left(e^{t/2}cos(W_t)\right)^2] & = \int_0^t \mathbb{E}[\left(e^{t/2}cos(W_t)\right)^2] dt \\ \leq \int_0^t \mathbb{E}[\left(e^{t/2}\right)^2] dt & = e^{t/2}1. \\ \therefore \left(e^{t/2}\right)^2\mathbb{E}[\left(cos(\theta)\right)^2] & \leq e^{t/2}1 \\ \therefore \mathbb{E}[\left(cos(\theta)\right)^2] &\leq e^{t/2}(1e^{t/2}) \\ \end{align} Therefore we have that: \begin{align} Var(e^{t/2}cos(\theta))& = Var(e^{t/2}cos(W_tW_0)) = \mathbb{E}[\left(e^{t/2}cos(W_t)\right)^2]  \mathbb{E}[cos(\theta)]^2 \\ & =\mathbb{E}[\left(e^{t/2}cos(W_t)\right)^2] \left(e^{t/2}\right)^2 \\ & \leq e^{t/2}(1e^{t/2})  \left(e^{t/2}\right)^2. \end{align} Since the lowerbound of $e^{t/2}cos(W_t)$ is $0$ then we have the following estimate on the variance: \begin{align} \therefore 0 \leq Var(cos(\theta)) & \leq e^{t/2}\left(12e^{t/2}\right). \end{align}
Given that $x\sim \mathcal{N}(\mu,\sigma^2)$, I used Mathematica to explicitly compute the integral corresponding to the expectation of $\sin(ax)$ and $\cos(ax)$. This would generalize all of the previous responses already given.
\begin{align*} &\mathbb{E}\left \{ \sin(ax) \right \}=\sin( a\mu) \exp \left ( \dfrac{1}{2} a^2 \sigma^2 \right ). \\ & \mathbb{E}\left \{ \cos(ax) \right \}=\cos( a\mu) \exp \left ( \dfrac{1}{2} a^2 \sigma^2 \right ). \end{align*}
Putting $\mu=0$ and $a=1$ will recover the desired result.
For $y=\cos(x)$, the CDF of $y$
\begin{array}{l} F_Y \left( y \right) = \left\{ {\begin{array}{*{20}c} {0,y <  1} \\ {P\left( {2k\pi + \arccos y \le x \le 2\left( {k + 1} \right)\pi  \arccos y} \right),k \in ,  1 \le y \le 1} \\ {1,y > 1} \\ \end{array}} \right. \\ P\left( {2k\pi + \arccos y \le x \le 2\left( {k + 1} \right)\pi  \arccos y} \right) \\ = \sum\limits_{k =  \infty }^{ + \infty } {\int_{2k\pi + \arccos y}^{2\left( {k + 1} \right)\pi  \arccos y} {f_X (x)} } dx \\ f_Y (x) = \sum\limits_{k =  \infty }^{ + \infty } {\left[ {  \left( {  \frac{1}{{\sqrt {1  y^2 } }}} \right)f_X \left( {2\left( {k + 1} \right)\pi  \arccos y} \right)  \left( {  \frac{1}{{\sqrt {1  y^2 } }}} \right)f_X \left( {2k\pi + \arccos y} \right)} \right]} \\ = \frac{1}{{\sqrt {1  y^2 } }}\sum\limits_{k =  \infty }^{ + \infty } {\left[ {f_X \left( {2\left( {k + 1} \right)\pi  \arccos y} \right) + f_X \left( {2k\pi + \arccos y} \right)} \right]} \\ \end{array}