The value of \( \cos^{-1}(\cos(\frac{7\pi}{6})) \) is:

Since \( \frac{7\pi}{6} \notin [0, \pi] \), we use \( \cos(\frac{7\pi}{6}) = \cos(2\pi - \frac{7\pi}{6}) = \cos(\frac{5\pi}{6}) \). Since \( \frac{5\pi}{6} \in [0, \pi] \), the value is \( \frac{5\pi}{6} \).