The Matlab script that I used to find these relations is below.
%% Cody Martin% m-file 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 degreesfor theta = -pi/2:pi/36:pi/2 theta1sig = pi/36; % standard deviation of orinigal normal distribution vtheta = theta + theta1sig*randn(9999,1); % create 9999 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 9999 points plot(sort(vctheta),[1:length(vctheta)]/length(vctheta),'k','LineWidth',2); % plot cdf of cos(9999 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 9999','Cos(Norm CDF 9999)','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 9999','Sin(Norm CDF 9999)','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)];figure(2); clf;subplot(211); hold on;title('Standard Deviation of Cosine of a Normal Distribution as a Function of the Original Average');legend('From 9999 Points','Fit: std = |\sigmasin(\mu)|');ylabel('std(cos(\theta_{vector})) [rad]');xlabel('\theta [rad]');subplot(212); hold on;title('Standard Deviation of Sine of a Normal Distribution as a Function of the Original Average');legend('From 9999 Points','Fit: std = |\sigmacos(\mu)|');ylabel('std(sin(\theta_{vector})) [rad]');xlabel('\theta [rad]');
