Answer
$\frac{\sqrt3}{2}$
Work Step by Step
RECALL:
$\sin{s} = y
\\\cos{s} = x
\\\tan{s} = \frac{y}{x}
\\\cot{s} = \frac{x}{y}
\\\sec{s} = \frac{1}{x}
\\\csc{s}=\frac{1}{y}$
(refer to Figure 11 , page 111 of the textbook)
Moving $\frac{4\pi}{3}$ in counterclockwise (negative direction) yields the same terminal side as $\frac{2\pi}{3}$.
This angle intersects the unit circle at the point $(-\frac{1}{2}, \frac{\sqrt3}{2})$.
This point has:
$x= -\frac{1}{2}$
$y=\frac{\sqrt3}{2}$
Thus,
$\sin{(-\frac{4\pi}{3})}
\\= \sin{\frac{2\pi}{3}}
\\=\frac{\sqrt3}{2}$