The range of the inverse cosine function, \(\cos^{-1}(x)\), is \([0, \pi]\) because of how the cosine function behaves and the requirement that an inverse function must be a function (i.e., each input must correspond to exactly one output).

### Explanation:

1. **Cosine Function Behavior**:
   - The cosine function, \(\cos(\theta)\), is periodic with a period of \(2\pi\). This means it repeats its values over every interval of \(2\pi\).
   - Over one period, \(\cos(\theta)\) starts from \(\cos(0) = 1\), decreases to \(\cos(\pi) = -1\), and then returns to \(\cos(2\pi) = 1\).

2. **Choosing the Range**:
   - The function \(\cos(\theta)\) is **decreasing** on the interval \([0, \pi]\). This means that for every value of \(\cos(\theta)\) in this interval, there is a unique corresponding angle \(\theta\) in the same interval.
   - The interval \([0, \pi]\) covers the entire range of possible cosine values, from \(\cos(0) = 1\) to \(\cos(\pi) = -1\). No other interval does this while maintaining the uniqueness required for an inverse function.

3. **Inverse Function Requirement**:
   - For the inverse cosine function \(\cos^{-1}(x)\) to be well-defined, it must map every \(x \in [-1, 1]\) to a unique angle \(\theta\). This angle must be in a range where the cosine function is **one-to-one** (i.e., it doesn’t repeat any value).
   - The interval \([0, \pi]\) is the standard choice because it captures all possible values of \(\cos(\theta)\) exactly once.

Thus, the range of \(\cos^{-1}(x)\) is \([0, \pi]\), ensuring that each \(x\) in the domain \([-1, 1]\) corresponds to a unique angle.

Would you like further details or have any other questions?

Here are five related questions that you might find interesting:

1. What is the range of the inverse sine function, \(\sin^{-1}(x)\)?
2. Why is the range of the inverse tangent function different from \(\cos^{-1}(x)\)?
3. How does the periodic nature of trigonometric functions affect their inverses?
4. What happens if you try to evaluate \(\cos^{-1}(x)\) for a value outside the domain \([-1, 1]\)?
5. How does the range of \(\cos^{-1}(x)\) relate to the unit circle?

**Tip:** Understanding the behavior of trigonometric functions on their principal intervals is key to mastering their inverses. Always visualize or refer to the unit circle when working with these concepts!

Explore why the range of the inverse cosine function \(\cos^{-1}(x)\) is \([0, \pi]\) with detailed explanations. This problem delves into trigonometric concepts, inverse functions, and periodicity, crucial for Grades 10-12 students.

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