Why does 'x' need to be in radians to differentiate 'sin x'?

There are two definitions of the sine and cosine functions that anyone who uses contemporary maths, and I do mean anyone, uses silently or otherwise. The first is as follows:'Rotate the point (1,0) in Euclidean (x,y)-space through some angle 'α' counter-clockwise. The image of (1,0) through this rotation is (cos α, sin α).'We then extend this definition to have α in radians if we want to perform calculus with these functions; similar to how we used geometry to define these functions, we use geometry to deduce the requisite limits for deriving the first derivatives of these functions. That geometry can only be performed as standard where we take α in radians.
The second is the one virtually everyone, including you, use day to day. Moreover, the last definition implies a limited version of this second one. We define the sines and cosines as infinite power series of radians that take complex numbers to complex numbers. Incorporated into these definitions is the need for 'x' to be in radians and the standard result 'd/dx(sin x)=cos x'.Either way, you need 'x' to be in radians at best to be able to prove geometrically across real angles the standard results desired and at worst as good as 'because I said so'.