Answer
$-2$
Work Step by Step
$\dfrac{2\pi}{3}$ is an angle whose terminal side is in Quadrant II.
The reference angle of an angle $\theta$ in Quadrant II can be found using the formula $\pi - \theta$.
Thus, the reference angle of the given angle is:
$=\pi - \dfrac{2\pi}{3}
\\=\dfrac{3\pi}{3} -\dfrac{2\pi}{3}
\\=\dfrac{\pi}{3}$
With a reference angle of $\dfrac{\pi}{3}$, then the cosine value of $\dfrac{2\pi}{3}$ will be the same as $\dfrac{\pi}{3}$ except possibly for its sign.
The value of cosine of $\dfrac{\pi}{3}$ is $\dfrac{1}{2}$. However, since $\dfrac{2\pi}{3}$ is in Quadrant II, its cosiine is negative.
Thus, $\cos{(\frac{2\pi}{3})} = -\dfrac{1}{2}$.
RECALL:
$\sec{\theta} = \dfrac{1}{\cos{\theta}}$
Using the formula above gives:
$\sec{(\frac{2\pi}{3})}=\dfrac{1}{\cos{(\frac{2\pi}{3})}}=\dfrac{1}{-\frac{1}{2}}=1 \cdot \dfrac{-2}{1} = -2$
