Answer
$$2$$
Work Step by Step
$$\eqalign{
& \csc \left( { - \frac{{11\pi }}{6}} \right) \cr
& {\text{Use the reciprocal identity }}\csc \theta = \frac{1}{{{\text{sin}}\theta }} \cr
& \csc \left( { - \frac{{11\pi }}{6}} \right) = \frac{1}{{\sin \left( { - 11\pi /6} \right)}} \cr
& {\text{Use sin}}\left( { - \theta } \right) = - \sin \theta \cr
& \csc \left( { - \frac{{11\pi }}{6}} \right) = \frac{1}{{ - \sin \left( {11\pi /6} \right)}} \cr
& {\text{Write }}\frac{{11\pi }}{6}{\text{ as 2}}\pi - \frac{\pi }{6} \cr
& \csc \left( { - \frac{{11\pi }}{6}} \right) = \frac{1}{{ - \sin \left( {{\text{2}}\pi - \frac{\pi }{6}} \right)}} \cr
& \csc \left( { - \frac{{11\pi }}{6}} \right) = \frac{1}{{ - \sin \left( { - \frac{\pi }{6}} \right)}} \cr
& \csc \left( { - \frac{{11\pi }}{6}} \right) = \frac{1}{{\sin \left( {\frac{\pi }{6}} \right)}} \cr
& {\text{From the unit circle we known that }}\sin \left( {\frac{\pi }{6}} \right) = \frac{1}{2} \cr
& \csc \left( { - \frac{{11\pi }}{6}} \right) = 2 \cr} $$
