Why are radians used in calculus. Ok, please ignore my silliness. 
So, why do we use radians in calculus and why is it considered more scientific than degrees. And how did mathematicians know or prove that radians would work for all integration, differentiation in other fields or is there a method to prove it?
 A: If degrees are used then
$$
\frac d{dx} \sin x = \frac \pi {180} \cos x.
$$
If some units other than degrees are used then
$$
\frac d{dx} \sin x = \left(\text{some constant}\cdot\cos x\right).
$$
If the units are radians, then the "constant" is $1$, so $\dfrac d{dx}\sin x = \cos x$.
As for proving this, go back to the proof that
$$
\lim_{x\to0} \frac{\sin x} x = \text{something}.
$$
The "something" is $\dfrac\pi{180}$ if degrees are used, and $1$ if radians are used, and something else if some other units are used.  Read that proof, and read the proof that $\sin'=\cos$, and notice how the proof of the former proposition is used in proving the latter proposition.
It's really the same as what's "natural" about the number $e$.
$$
\frac d{dx} 2^x = (\text{some constant}\cdot2^x).
$$
$$
\frac d{dx} 10^x = (\text{some other constant}\cdot10^x).
$$
$$
\frac d{dx} e^x = (\text{some constant}\cdot e^x).
$$
Only when the base is $e$ is the "constant" equal to $1$.
A: Radians make many formulas much simpler.
For example, the length of an arc of a circle subtended by an angle of $\theta^\circ$
$$s=2\pi r\theta/360^\circ.$$
If $\theta$ is measured in radians: we get $s=r\theta$.
Or, the area of a sector is 
$$A=\pi r^2\theta/360^\circ,\quad\theta\text{ in degrees}$$
but
$$A=r^2\theta/2\quad\theta\text{ in radians}.$$
It makes sense, that the formulas for derivatives, and antiderivatives would be much nicer in radians.
