Find the principal value of .

Let $\theta = \sec^{-1}(-\sqrt{2})$. Then $\sec \theta = -\sqrt{2}$, which means $\cos \theta = -\frac{1}{\sqrt{2}}$. The principal value branch of $\sec^{-1}$ is $[0, \pi] \setminus \{\pi/2\}$. Since $\cos \theta < 0$, $\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$.