Answer
$\color{blue}{\sec{(\frac{2\pi}{3})}=-2}$
Work Step by Step
Figure 13 on page 579 of this book shows that the unit circle point $(-\frac{1}{2}, \frac{\sqrt3}{2})$ corresponds to the real number $\dfrac{2\pi}{3}$.
RECALL:
(i) $\cos{s} = x$
(ii) $\sin{s} = y$
(iii) $\tan{s}=\dfrac{y}{x}$
(iv) $\sec{s} =\dfrac{1}{x}$
Use the coordinates of the unit circle point above and the formula in (iv) above to obtain:
$\color{blue}{\sec{(\frac{2\pi}{3})}=\frac{1}{-\frac{1}{2}}=1 \cdot (-2)=-2}$
