Answer
$120^{o}$
Work Step by Step
Solve for radians, then convert to degrees.
$y=\sec^{-1}x =$arcsec$x$
Domain: $(-\infty, -1]\cup[1, \infty)$
Range: $[0,\displaystyle \frac{\pi}{2})\cup(\frac{\pi}{2},\pi]$
Quadrants (unit circle): I and II
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$y$ is the number from $[0,\displaystyle \frac{\pi}{2})\cup(\frac{\pi}{2},\pi]$
such that $\sec y=-2$
$(\displaystyle \cos y=\frac{1}{-2}=-\frac{1}{2})$
In quadrant I, $\displaystyle \cos\frac{\pi}{3}=\frac{1}{2}$.
In quadrant II, $\displaystyle \cos\frac{2\pi}{3}=-\frac{1}{2}$, that is,
$\displaystyle \sec\frac{2\pi}{3}=-2, $ and
$\displaystyle \frac{2\pi}{3}\in[0,\frac{\pi}{2})\cup(\frac{\pi}{2},\pi]$.
So,
$y=\displaystyle \frac{2\pi}{3}$
To convert radians to degrees, multiply y with $\displaystyle \frac{180^{o}}{\pi}$
$\displaystyle \theta=\frac{2\pi}{3}\cdot\frac{180^{o}}{\pi}=120^{o}$
