Which of the following of the principal value branch of `cos^(-1) x` ?

Options

A. `[-(pi)/(2), (pi)/(2)]`
B. `(0,pi)`
C. `[0,pi]`
D. `(0,pi)-{pi/2}`

Explanation

To determine the principal value branch of \( \cos^{-1} x \), we need to analyze the function and its range. Here’s a step-by-step solution:

### Step 1: Understanding the Function
The function \( \cos^{-1} x \) is the inverse of the cosine function. It gives us the angle whose cosine is \( x \).

### Step 2: Identifying the Domain
The domain of \( \cos^{-1} x \) is restricted to the interval \( [-1, 1] \) because the cosine function only takes values in this range.

### Step 3: Identifying the Range
The range of the principal value branch of \( \cos^{-1} x \) is the set of angles that the function can output. For the inverse cosine function, this is defined to be from \( 0 \) to \( \pi \).

### Step 4: Conclusion
Thus, the principal value branch of \( \cos^{-1} x \) is given by the interval:
\[ 0 \leq \cos^{-1} x \leq \pi \]

### Final Answer
The principal value branch of \( \cos^{-1} x \) is \( [0, \pi] \).

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