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

Explanation

To solve the problem of finding the value of \( \cos^{-1}(\cos 13) \), we can follow these steps:

### Step 1: Understand the Range of \( \cos^{-1} \)
The function \( \cos^{-1}(x) \) (also known as arccosine) has a range of \( [0, \pi] \). This means that the output of \( \cos^{-1}(x) \) will always be between 0 and \( \pi \).

### Step 2: Check the Argument
The argument in our case is \( 13 \), which is outside the range of \( [0, \pi] \) since \( 13 \) is greater than \( \pi \) (approximately 3.14).

### Step 3: Use the Periodicity of Cosine
We know that cosine is periodic with a period of \( 2\pi \). Therefore, we can express \( 13 \) in terms of \( 2\pi \):
\[ 13 = 2\pi k + \theta \]
where \( k \) is an integer and \( \theta \) is the angle that lies within the range of \( [0, 2\pi) \).

### Step 4: Find the Equivalent Angle
To find the equivalent angle that lies within \( [0, 2\pi) \), we can subtract \( 2\pi \) from \( 13 \):
\[ 2\pi \approx 6.28 \quad \Rightarrow \quad 13 - 2\pi \approx 13 - 6.28 \approx 6.72 \]
Since \( 6.72 \) is still greater than \( \pi \), we subtract \( 2\pi \) again:
\[ 6.72 - 2\pi \approx 6.72 - 6.28 \approx 0.44 \]
Now, \( 0.44 \) is within the range of \( [0, \pi] \).

### Step 5: Write the Final Expression
Now, we can express \( \cos^{-1}(\cos 13) \) as:
\[ \cos^{-1}(\cos 13) = \cos^{-1}(\cos(0.44)) \]
Since \( 0.44 \) is within the range of \( [0, \pi] \), we have:
\[ \cos^{-1}(\cos(0.44)) = 0.44 \]

### Final Answer
Thus, the value of \( \cos^{-1}(\cos 13) \) is:
\[ \boxed{0.44} \]