I have a question on angles expressed in term of radian versus angles expressed in term of degree, which I would like to elevate them into abstract algebra:
(1) Why engineering math uses radian instead of degree? Judging from the way Calculus textbooks very casually write expressions such as these:
$$\int (\theta^2 + \sec^2 \theta) d \theta, \ \int (\cos x + 3^x) dx, \ \int (4x - \csc^2x) dx,$$
do you think it is because angle stated in radian, unlike in degree, can be treated as scalar and be multiplied with each other, since radian actually comes from arc length?
(2) If that is true, do you think that the set of angles expressed in radian, $\mathcal R$, and the set of angles expressed in degree, $\mathcal D$, are probably good examples of ring and group, in that $\mathcal D$ takes only addition as the only operation?
(3) Taking it one more step further down, do you think $\mathcal D$ is also a good example of $\mathbb R$-module?
I am asking these questions not only for my curiosity, but also exploring the possibility of using them in my master thesis, although I am not sure if they have enough depth and breadth. Thank you all for your time and effort.
POST SCRIPT: I think my question is different from those linked to this posting. Whereas the others asked mostly "Why calculus uses radian instead of degree?" but I asked "It is true that radian takes multiplication and degree doesn't?" since I would like to take them as example of group versus ring. Having said that, however, from the multitude of links that I receive here, it appears that my observation is wrong. Calculus uses radian for some other deeper reasons. Thank you all very much for all your time and help. Happy Halloween.
