Chapter 6 - Analytic Trigonometry - Section 6.2 The Inverse Trigonometric Functions (Continued) - 6.2 Assess Your Understanding - Page 480: 42

$\dfrac{2 \pi}{3}$

We know that the trigonometric function $\sec^{-1} x$ has range: $[0, \dfrac{\pi}{2} ) \cup ( \dfrac{\pi}{2}, \pi]$. Here, $\sec^{-1} (-2)=\dfrac{2 \pi}{ 3}$, because $\sec(\dfrac{2\pi}{3})=-2$ and $\dfrac{2 \pi}{3}$ lies in the range of $\sec^{-1}{x}$ .