The principal values of cos^-1≤ft(-sin(7π)/6), is

We have

$cos^{-1}\left(-sin\frac{7\pi}{6}\right)= cos^{-1}\left(-sin\left(\pi +\frac{\pi}{6}\right)\right) = cos^{-1}\left(sin\frac{\pi}{6}\right)$

$=\frac{\pi}{2}-sin^{-1}\left(sin\frac{\pi}{6}\right)=\frac{\pi}{2}-\frac{\pi}{6}=\frac{\pi}{3}$