Find the value the following :
` cos^(-1) (cos . (13pi)/6)`

Answer

To find the value of \( \cos^{-1} \left( \cos \left( \frac{13\pi}{6} \right) \right) \), we will follow these steps:

### Step 1: Simplify the angle
First, we need to simplify the angle \( \frac{13\pi}{6} \). We can do this by finding an equivalent angle within the range of \( [0, 2\pi] \).

\[ \frac{13\pi}{6} - 2\pi = \frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6} \]

So, \( \frac{13\pi}{6} \) is equivalent to \( \frac{\pi}{6} \) in the range of \( [0, 2\pi] \).

### Step 2: Evaluate the cosine
Now, we can evaluate \( \cos \left( \frac{13\pi}{6} \right) \):

\[ \cos \left( \frac{13\pi}{6} \right) = \cos \left( \frac{\pi}{6} \right) \]

From trigonometric values, we know:

\[ \cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} \]

### Step 3: Apply the inverse cosine function
Next, we need to find \( \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) \). The inverse cosine function gives us the angle whose cosine is \( \frac{\sqrt{3}}{2} \).

The angle in the range \( [0, \pi] \) that corresponds to \( \frac{\sqrt{3}}{2} \) is:

\[ \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) = \frac{\pi}{6} \]

### Final Answer
Thus, the value of \( \cos^{-1} \left( \cos \left( \frac{13\pi}{6} \right) \right) \) is:

\[ \frac{\pi}{6} \]

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