1. **Problem Statement:** We want to define the inverse cosine function, denoted as $\arccos(x)$, which requires restricting the domain of the cosine function to an interval where it is one-to-one (injective) and thus invertible. 2. **Periodicity and Injectivity:** The cosine function $\cos(x)$ is periodic with fundamental period $2\pi$, meaning $\cos(x) = \cos(x + 2\pi k)$ for any integer $k$. 3. **Domain Restriction:** To have an inverse, we restrict the domain to an interval of length $\pi$ where $\cos(x)$ is strictly decreasing and injective. By convention, this interval is $[0, \pi]$. 4. **Why $[0, \pi]$?** On $[0, \pi]$, $\cos(x)$ decreases from 1 to -1 without repeating values, making it invertible. 5. **Inverse Function Definition:** The inverse cosine function $\arccos(y)$ is defined as the unique $x \in [0, \pi]$ such that $\cos(x) = y$ for $y \in [-1,1]$. 6. **Summary:** - Cosine is periodic with period $2\pi$. - Restrict domain to $[0, \pi]$ for invertibility. - $\arccos$ maps $[-1,1]$ back to $[0, \pi]$. This restriction ensures the inverse cosine function is well-defined and consistent with the standard convention.