How were trigonometrical functions and its inverses discovered?

Question

Imagine you just did a circle. Some functions are just definitions, like $\sin$,$\cos$ and $\tan$ but how do you derive a formula to get the $\sin$ from an angle in radians (maybe by Taylor series, but then you should know that $(\sin(x))'=\cos(x)$)? Or a formula to get the original angle from its $\sin$?