For a measurement $\theta$ with uncertainty $\Delta\theta$, the propagated error of the sine function is given by
$$\Delta(\sin\theta) = (\cos\theta)\cdot\Delta\theta$$
However I note that if $\theta$ is given in terms of radians and degrees, $\Delta\theta$ will be scaled accordingly.
i.e.
$$\Delta(\sin\theta_{radian}) = (\cos\theta_{radians})\cdot\Delta\theta_{radians}$$
$$\Delta(\sin\theta_{degrees}) = (\cos\theta_{degrees})\cdot\Delta\theta_{degrees}$$
Which gives different values of error for $\Delta(\sin\theta)$
How should the error of propagation be evaluated? Is the convention for it to be evaluated in radians?
Best Answer
What you are doing is approximating $f(x+h)-f(x) \approx f'(x)h$.
If $f(x) = \sin x$ then $f(x+h)-f(x) \approx (\cos x) h$.
If $f(x) = \sin ({\pi \over 180} x)$, then $f(x+h)-f(x) \approx {\pi \over 180}(\cos ({\pi \over 180} x)) h$.
The third formula in the question is incorrect.