Arccosine Function

Approach: We need an angle $\theta \in [0, \pi]$ with $\cos \theta = -\frac{1}{2}$.

Steps:

1. Among the special angles, $\cos \frac{2\pi}{3} = -\frac{1}{2}$;
2. $\frac{2\pi}{3} \approx 2.094$, which lies in $[0, \pi]$;
3. By uniqueness on the principal interval, $\arccos(-\frac{1}{2}) = \frac{2\pi}{3}$.

Answer: $\arccos(-\frac{1}{2}) = \frac{2\pi}{3}$.

Approach:

The arccosine is the inverse of the cosine function restricted to its principal interval $[0, \pi]$: the domain comes from the range of $\cos x$, and the range is the principal interval itself.

Steps:

1. The range of the cosine function $y = \cos x$ on $\mathbb{R}$ is $[-1, 1]$, and the domain of an inverse function is exactly the range of the original function, so the domain of $y = \arccos x$ is $[-1, 1]$.
2. The cosine function is monotonically decreasing on $[0, \pi]$, running from $1$ down to $-1$, so this interval is chosen as the principal interval to make the inverse single-valued.
3. The inverse maps the angles of the principal interval back onto the domain of the original function, so the range of $y = \arccos x$ is $[0, \pi]$.

Answer:

Domain: $[-1, 1]$; Range: $[0, \pi]$.