I wonder why uncertainties in angle measurement MUST be in radians. For example, I want to calculate the uncertainty in measuring the function $y= \sin (\theta)$ when the angle is measured $\theta = 63$ $\pm 1$ degree. I do this using differential, so $dy = \cos (\theta) d\theta$, now $d\theta = \pm 1$ degree is the error in $\theta$. Now, all the course notes/ books I read says this must be converted in radians, even though the angle we use here is measured in degree. How come?
Thanks Cal2
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They don't have to be. It is just if you don't you will have a bunch of $\frac{\pi}{180}$ factors when you differentiate $\sin \frac{\pi x}{180}$ which is the expression for $\sin$ in degrees. You could use whatever coordinate you want, but it would be silly to not use the one that makes the computation simpler.
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It is not exactly true that the error in measuring an angle "must be in radians" even if the degrees? The error of any measurement must be given in the same units as the measurement itself.
HOWEVER, you are not asking about the error in measuring an angle, you are asking about the error in the value of a function of the angle. To do that, you are using the fact that if y= sin(x) then dy= cos(x) dx which is, as Martin Argerami said, true only as long as x itself is measured in radians. Of course, dy, the error in this function would not be measured in radians- y is not an angle at all.
(One can show that, if y= sin(x) and x is measured in degrees, then $dy= \frac{\pi}{180}cos(x)dx$. That will give the same result as changing x to radians and using $dy= cos(x)dx$.)
Because, if you don't use radians the derivative of $\sin\theta$ is not $\cos\theta$, and so your formula $dy=\cos\theta\,d\theta$ doesn't hold (it needs a coefficient).