Propagating uncertainty in trigonometric functions for angle measurements

[fs93]

Hello fellow physicists :)

I have recently done an experiment and am trying to propagate the uncertainty. Unfortuantely, I haven't done that in years, and need to remember how its done.
For example:

When theta= 20 (+-2)

I want to find sin20 and the uncertainty:

sin20=0.34 (+-?)

How can I find the uncertainty? Do I do this?

sin18=0.31

sin 22=0.37

And subsequently : sin20=0.34 (+-0.03)?

Is this method correct? If not please advise me as to how I should propagate the uncertainties.

Thanks in advance,

FS

 

[jtbell]

Yes, that's a valid method. I use it often myself.

 

[elduderino]

If you want a rigorously mathematical and more accurate approach, I would recommend using the Taylor Series.

suppose you know the uncertainty in x to be $$\delta x$$
you use the taylor expansion of f( x+$$\delta x$$ ), ignoring terms $$O(\delta x^n)$$ and higher, you would get the uncertainty in your function(n would depend on the accuracy you need). In your case the variable is theta and the function is the sine function.

suppose you are calculating the error in the neighbourhood of "a":

$$f(a)+\frac {f'(a)}{1!} \delta x + \frac{f''(a)}{2!} \delta x^2+\frac{f^{(3)}(a)}{3!}(\delta x^3)+ \cdots$$

 

[fs93]

Thanks both, I was looking for something more like Eldudrino's equation and I will be using it.

Cheers!

FS

 

[f95toli]

If you want a rigorously mathematical and more accurate approach

Actually, if you want to be very rigourous you should be using whatever mathod is recommended in GUM for you particual situation since you are then following the international standard (you should be a be able to find the GUM as a PDF file if you google ISO GUM, I think there is even a wiki).

GUM is actually quite good as a "howto" manual for cases like this.