The Inverse Cosine Function and its Properties

Cosine is not one-to-one over $\mathbb{R}$, but restricting it to $[0,\pi]$ makes it one-to-one, still with range $[-1,1]$.

Definition 4.4. For $-1\le x\le1$, $\cos^{-1}x$ is the UNIQUE number $y\in[0,\pi]$ such that $\cos y=x$. In symbols, $\cos^{-1}:[-1,1]\to[0,\pi]$ is defined by $\cos^{-1}(x)=y\iff\cos y=x$ and $y\in[0,\pi]$.

Notes.

1. Sine is non-negative on $[0,\pi]$ — the range of $\cos^{-1}x$ — again relevant to later trigonometric substitutions.
2. $\cos:[0,\pi]\to[-1,1]$ and $\cos^{-1}:[-1,1]\to[0,\pi]$.
3. Cosine could also be restricted to $[-\pi,0]$ or $[\pi,2\pi]$ and remain one-to-one with range $[-1,1]$, but $[0,\pi]$ is the CHOSEN principal domain. $[0,\pi]$ is the principal domain of cosine, and the values of $y=\cos^{-1}x$ are the principal values. From the definition:

(i) $y=\cos^{-1}x\iff x=\cos y$, for $-1\le x\le1$ and $0\le y\le\pi$.

(ii) $\cos(\cos^{-1}x)=x$ if $x\le1$, and is meaningless if $x>1$.

(iii) $\cos^{-1}(\cos x)=x$ if $0\le x\le\pi$ — the range of $\cos^{-1}x$. NOTE: $\cos^{-1}\left(\cos\tfrac{3\pi}2\right)=\tfrac{\pi}2\ne\tfrac{3\pi}2$. …