Answer
$-2$
Work Step by Step
RECALL:
In a unit circle
$\cos {t}=x
\\\sin{t}=y
\\\tan{t} = \dfrac{y}{x}, x\ne0
\\\cot{t} = \dfrac{x}{y}, y \ne 0
\\\sec{t} = \dfrac{1}{x}, x\ne0
\\\csc{t} = \dfrac{1}{y}, y \ne0$
Using the definition above and the unit circle in Figure 5 on page 137 of this book, then:
Point on the unit circle: $\left(-\dfrac{\sqrt3}{2}, -\dfrac{1}{2}\right)$
Thus,
$\csc{210^o} = \dfrac{1}{y} = \dfrac{1}{-\frac{1}{2}}=1 \cdot \dfrac{-2}{1} = -2$