Answer
$$ - \frac{1}{2}$$
Work Step by Step
$$\eqalign{
& \sin \left( { - \frac{{5\pi }}{6}} \right) \cr
& = - \sin \left( {\frac{{5\pi }}{6}} \right) \cr
& {\text{Write }}\frac{{5\pi }}{6}{\text{ as }}\pi - \frac{\pi }{6} \cr
& \sin \left( { - \frac{{5\pi }}{6}} \right) = - \sin \left( {\pi - \frac{\pi }{6}} \right) \cr
& {\text{Use the identity sin}}\left( {\pi - \theta } \right) = + \sin \theta \cr
& \sin \left( { - \frac{{5\pi }}{6}} \right) = - \sin \left( {\frac{\pi }{6}} \right) \cr
& {\text{From the unit circle we known that sin}}\frac{\pi }{6} = \frac{1}{2},{\text{ then}} \cr
& \sin \left( { - \frac{{5\pi }}{6}} \right) = - \frac{1}{2} \cr} $$
