Inverse Cosine Function

Consider $\cos$ restricted to $[0,\pi]\to[-1,1]$. Graphically this restriction is one-one (strictly decreasing) and onto $[-1,1]$, so its inverse exists, called the inverse cosine function, denoted $\cos^{-1}:[-1,1]\to[0,\pi]$. For $x\in[-1,1]$ and $\theta\in[0,\pi]$, we write $\cos^{-1}x=\theta$ if $\cos\theta=x$; $\theta$ is the principal value of $\cos^{-1}x$.

Ex. $\cos\dfrac{\pi}{4}=\dfrac{1}{\sqrt2}$, where $\dfrac{1}{\sqrt2}\in[-1,1]$ and $\dfrac{\pi}{4}\in[0,\pi]$, so $\cos^{-1}\dfrac{1}{\sqrt2}=\dfrac{\pi}{4}$: the principal value of $\cos^{-1}\dfrac{1}{\sqrt2}$ is $\dfrac{\pi}{4}$. Although $\cos\left(-\dfrac{\pi}{4}\right)=\dfrac{1}{\sqrt2}$ also, we cannot write $\cos^{-1}\dfrac{1}{\sqrt2}=-\dfrac{\pi}{4}$, since $-\dfrac{\pi}{4}\notin[0,\pi]$. …