Question

why are the derivatives of trigonometric functions measured in radians and not degrees?

Answer 1

Try to play with http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html or WolframAlpha, or even better Mathematica if you you can get your hands on that software.

There is a difference in the evaluation of the two. In general think of a sine as a function defined between \(\mathbb{R} \to \mathbb{R}\) in general you can remember that \[°= \times \frac{\pi}{180} \]