What's so special about radians? (Differentiation) It seems to me that radians have lots of very special properties that allow us to do maths with trigonometric functions. When I first came across radians, I was led to believe that they were designed purely to make finding areas and arc lengths of sectors more easy. But then I discovered that:
$$\frac{d}{dx}\sin x=\cos x$$
Clearly this only works when using radians. If you used degrees and found the gradient of $\sin x$ at $x=0$ you would get  that gradient $=\frac{\pi}{180}^{\circ}$. So my question is: why does differentiating $\sin x$ give you $cosx$ when you are using radians? What makes radians so special in this respect?
 A: No matter which unit for angles you use, the derivative of the sine will be some multiple of the cosine.
The radian measure is chosen as the angle measure that makes the proportionality constant $1$. If it was a different angle measure that did this we would be working with that as the natural angle measure instead of radians.
Geometrically think of the sine of a very small angle, that is, the ratio between short side and the hypotenuse in a very thin right triangle. We can scale everything so the hypotenuse has length 1; then the sine is simply the length of the short side.
So if we want $\sin'(0)=1\cdot\cos(0) = 1$ we had better choose our angle measure such that the measure of a small angle is close to the length of that short side.
Now, if we draw a circular arc next to the short side with the hypotenuse as the radius, until it reaches the extension of the long leg of the triangle, this will almost coincide with the short side -- and the thinner the triangle is, the better does it match.
And "arc length of a unit circle" is clearly additive on angles, so it will work as an angle measure. Therefore it is the angle measure that makes the derivative of the sine at 0 unity.
A: to prove the differential of sin(x) is achieved using the fact that $$\lim_{x \rightarrow 0} \frac{sin(x)}{x} = 1$$
to prove that we say that for a small enough absoulte(x) $$sin(x)\leq x \leq tan(x)$$
this is proved using a bit (complicated) geometry, that is only true assuming x is in radians. 
KH is sin(x)
the arc KA is x
and LA is tan(x)
this however is not a proof,
the full proof is a bit more complicated.
