Find the principal values of the following
`cos^(-1) (cos. (13pi)/6)`

Answer

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

### Step 1: Simplify the angle
First, we need to simplify the angle \( \frac{13\pi}{6} \) to find an equivalent angle within the range of \( [0, 2\pi) \).

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

### Step 2: Use the property of the cosine function
Next, we can use the property of the cosine function, which states that \( \cos^{-1}(\cos \theta) = \theta \) when \( \theta \) is in the range \( [0, \pi] \).

Since \( \frac{\pi}{6} \) is within the range \( [0, \pi] \), we can directly apply this property.

### Step 3: Find the principal value
Now, we can find the principal value:

\[ \cos^{-1} \left( \cos \left( \frac{13\pi}{6} \right) \right) = \frac{\pi}{6} \]

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

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

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