Inverse Trigonometric Functions

So far we have treated $\sin$, $\cos$, $\tan$, etc. as functions that take an angle and produce a number. This section asks the reverse question: given the number, can we recover the angle? Recall that a function $f:A\to B$ has an inverse $f^{-1}:B\to A$ only when $f$ is one-one and onto; for $y=f(x)$, the inverse says $x=f^{-1}(y)$, and $\text{domain}(f^{-1})=\text{range}(f)$, $\text{range}(f^{-1})=\text{domain}(f)$. But no trigonometric function is one-one on its full domain — e.g. $\sin\theta=k$ (for $|k|<1$) already has infinitely many solutions $\theta=n\pi+(-1)^n\alpha$ (Theorem 3.1), so infinitely many angles share one sine value, and $\sin$ cannot be inverted on all of $\mathbb{R}$. However, examining the graph of each trigonometric function shows that on a suitably chosen restricted interval of its domain, it is one-one and onto — and it is only on that restricted int …