Answer
$\color{blue}{\sin{(-\frac{5\pi}{6})}=-\frac{1}{2}}$
Work Step by Step
Note that $-\dfrac{5\pi}{6}$ is coterminal with:
$-\dfrac{5\pi}{6} + 2\pi=-\dfrac{5\pi}{3} + \dfrac{12\pi}{6}=\dfrac{7\pi}{6}$
Figure 13 on page 579 of this book shows that the unit circle point $(-\frac{\sqrt3}{2}, -\frac{1}{2})$ corresponds to the real number $\dfrac{-5\pi}{6}$ or $\dfrac{7\pi}{6}$.
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:
$\color{blue}{\sin{(-\frac{5\pi}{6})}=\sin{\frac{7\pi}{6}}=-\frac{1}{2}}$
