T3 If we define the inverse sine of x as
then the inverse sine is defined for x between 0 and 1. Since the inverse sine is the integral of a positive function, it is increasing, and is, hence, one to one. Since it is one to one, it has an inverse. We define the sine to be the inverse of the inverse sine.
If we take the derivative of the inverse sine, we get
If we differentiate both sides of this equation,
If we define
then we get that the derivative of the sine is the cosine. It is another standard calculus problem at this point to get that the derivative of the cosine will be the negative of the sine
This gives us the derivatives of the sine and cosine long before we obtain similar triangles and the fact that there are 180o in a triangle which we would need to develop trigonometry.
Of course these functions are only defined for angles in the first quadrant at this point, but it is very standard to extend these definitions to more general angles.
Analytic Foundations of Geometry