I am currently writing a report and need to do the error propagation for a value. Here Bragg's law is changed so λ can be determined. An error was assumed to be 0.1° here. The question is how do I setup the equation for the error propagation here? Because I remember that for trigonometric functions degrees have to be changed to radians. But I don't know if this is necessary if I don't determine another angle here.
For example if I had sin(x) as function and then said, that in my experiment I got sin(0°) but my error might 2°, then my error propagation would be 2°x cos(0°) and that would be 2, so bigger than the actual range of a sine or cosine. But in my case, the error can be bigger than 1 since I am not looking at an angle anymore. Therefore, is it okay to use degrees here?
And the second question, as there will also be an error for an angle later in the report. If I need to switch to radians, how do I enter that in a calculator? Do I need to convert the error degrees, which are not part of a trigonometric function anymore to radians before (like I did on the picture) but leave the angle in the cosine function as degrees and set my calculator to do radians (so the 4th line in my picture)? Because otherwise the degrees outside the function would never be changed to radians in this process.
Best Answer
The general equation for a function $f(x,y) $ is to calculate the error on the result you want wrt the various terms in the equation, for example for equation that has errors in x and y, then the square of the total error (standard deviation) is $\sigma_f^2= [df/dx ]^2\sigma_x^2 + [df/dy]^2\sigma_y^2$, where the $\sigma$ are the errors, in your case on $\theta$ and d. Use radians not degrees in trig functions, also for errors (180 deg = $\pi$ radians). Thus in your example if the only error is in $\theta$, $\sigma_\lambda =2d \sqrt{sin(\theta)^2}\sigma_\theta$.