Answer
$\color{blue}{\sin{(-\frac{4\pi}{3})}=\dfrac{\sqrt3}{2}}$
Work Step by Step
Note that $-\dfrac{4\pi}{3}$ is coterminal with:
$-\dfrac{4\pi}{3} + 2\pi=-\dfrac{4\pi}{3} + \dfrac{6\pi}{3}=\dfrac{2\pi}{3}$
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 $\dfrac{-4\pi}{3}$ or $\dfrac{2\pi}{3}$.
RECALL:
(i) $\cos{s} = x$
(ii) $\sin{s} = y$
(iii) $\tan{s}=\dfrac{y}{x}$
(iv) $\sec{s} =\dfrac{1}{x}$
(v) $\csc{s} = \dfrac{1}{y}$
(vi) $\cot{s} = \dfrac{x}{y}$
Use the coordinates of the unit circle point above and the formula in (ii) above to obtain:
$\color{blue}{\sin{(-\frac{4\pi}{3})}=\sin{\frac{2\pi}{3}}=\frac{\sqrt3}{2}}$
