We are now going to consider the problem of finding the derivatives of the inverses of trigonometric functions. Now is a very good time to go back and reread Section 0.6 on inverse functions — especially Definition 0.6.4. Most importantly, given a function $f(x)\text{,}$ its inverse function $f^{-1}(x)$ only exists, with domain $D\text{,}$ when $f(x)$ passes the “horizontal line test”, which says that for each $Y$ in $D$ the horizontal line $y=Y$ intersects the graph $y=f(x)$ exactly once. (That is, $f(x)$ is a one-to-one function.)
Let us start by playing with the sine function and determine how to restrict the domain of $\sin x$ so that its inverse function exists.