1. The problem asks for the interval of the principal value of the function $\cos^{-1} x$ (arccosine of $x$) and to draw its graph. 2. The principal value of the inverse cosine function $\cos^{-1} x$ is defined as the angle $\theta$ such that $\cos \theta = x$ and $\theta$ lies in the interval $[0, \pi]$. 3. Therefore, the domain of $\cos^{-1} x$ is $[-1, 1]$ because cosine values range between $-1$ and $1$. 4. The range (interval of principal values) of $\cos^{-1} x$ is $[0, \pi]$. 5. The graph of $y = \cos^{-1} x$ starts at $( -1, \pi )$, decreases continuously, and ends at $(1, 0)$. 6. This function is decreasing and continuous on the domain $[-1, 1]$. Final answer: The interval for the principal value of $\cos^{-1} x$ is $[0, \pi]$.