It is not exactly clear what are trying to ask.
What ratio do you want the amplitude to be with respect to half the period. 1:180 or 1: i
Presuming want the former than:
y' = pi/180 * cos(x)
This more complicated than if ratio is 1: i. As if you let pi radians = 180 degrees, you will end up with y' = cos(x).
The proof of the theorem sin(x) d/dx = cos(x) is beyond the scope of advanced maths so just accept the theorem.
