Answer
$-\frac{1}{2}$
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 $\frac{5\pi}{6}$ in counterclockwise (negative direction) yields the same terminal side as $\frac{7\pi}{6}$.
This angle intersects the unit circle at the point $(-\frac{\sqrt3}{2}, -\frac{1}{2})$.
This point has:
$x= -\frac{\sqrt3}{2}$
$y=-\frac{1}{2}$
Thus,
$\sin{(-\frac{5\pi}{6})}
\\= \sin{\frac{7\pi}{6}}
\\=-\frac{1}{2}$