Find the value of : `cos^(-1)(cos 4)`

Explanation

To find the value of \( \cos^{-1}(\cos 4) \), we need to consider the properties of the cosine function and its inverse.

### Step-by-Step Solution:

1. **Understand the Range of \( \cos^{-1} \)**:
The function \( \cos^{-1}(x) \) (also known as arccos) has a range of \( [0, \pi] \). This means that any input to this function must yield an output that lies within this interval.

2. **Check the Value of 4**:
The value \( 4 \) is greater than \( \pi \) (approximately \( 3.14 \)). Therefore, \( \cos(4) \) will yield a value, but \( 4 \) itself is not in the range of \( \cos^{-1} \).

3. **Use the Periodicity of Cosine**:
Since cosine is periodic with a period of \( 2\pi \), we can express \( 4 \) in terms of an equivalent angle within the range of \( [0, 2\pi] \). We can subtract \( 2\pi \) from \( 4 \):
\[ 4 - 2\pi \approx 4 - 6.28 = -2.28 \]
However, since we want a positive equivalent angle, we can also consider:
\[ 2\pi - 4 \]

4. **Calculate \( 2\pi - 4 \)**:
Now, let's calculate \( 2\pi - 4 \):
\[ 2\pi \approx 6.28 \quad \Rightarrow \quad 2\pi - 4 \approx 6.28 - 4 = 2.28 \]

5. **Apply the Inverse Cosine**:
Now that we have \( 2\pi - 4 \) which is approximately \( 2.28 \), we can substitute this back into the inverse cosine function:
\[ \cos^{-1}(\cos 4) = 2\pi - 4 \]

6. **Final Answer**:
Thus, the value of \( \cos^{-1}(\cos 4) \) is:
\[ \boxed{2\pi - 4} \]