Find the value of ` cos ^(-1) ( cos 13) `.

Answer

To find the value of \( \cos^{-1}(\cos 13) \), we need to follow a few steps to ensure that we are working within the correct range for the inverse cosine function.

### Step 1: Understand the range of the inverse cosine function
The function \( \cos^{-1}(x) \) is defined for \( x \) in the range of \([-1, 1]\), and its output (the angle) lies in the range \([0, \pi]\).

### Step 2: Find the equivalent angle
Since \( 13 \) is outside the range \([0, \pi]\), we need to find an equivalent angle that lies within this range. We can do this by subtracting multiples of \( 2\pi \) (or \( 360^\circ \)) from \( 13 \).

### Step 3: Calculate \( 13 - 2\pi \)
First, we calculate \( 2\pi \):
\[ 2\pi \approx 6.2832 \]
Now, we subtract this from \( 13 \):
\[ 13 - 2\pi \approx 13 - 6.2832 \approx 6.7168 \]
Since \( 6.7168 \) is still greater than \( \pi \), we subtract \( 2\pi \) again:
\[ 6.7168 - 2\pi \approx 6.7168 - 6.2832 \approx 0.4336 \]
Now, \( 0.4336 \) is within the range \([0, \pi]\).

### Step 4: Use the property of inverse cosine
Now we can use the property:
\[ \cos^{-1}(\cos \theta) = \theta \quad \text{if } \theta \in [0, \pi] \]
Since \( 0.4336 \) is in the range, we have:
\[ \cos^{-1}(\cos 13) = 0.4336 \]

### Final Answer
Thus, the value of \( \cos^{-1}(\cos 13) \) is approximately \( 0.4336 \).