Answer
$\dfrac{2\sqrt3}{3}$
Work Step by Step
Convert the angle measure to degrees to obtain:
$=-\frac{13\pi}{3} \cdot \frac{180^o}{\pi} = -13(60)^o=-780^o$
Thus,
$\csc{(-\frac{13\pi}{3})} = \csc{(-780^o)}$
$-780^o$ is co-terminal with $-780^o+1080^o=300^o$.
$300^o$ is in Quadrant IV so its reference angle is $=360^o-300^o=60^o$.
Note that the cosecant function is positive in Quadrant IV.
From Section 2.1 (page 50) , we learned that:
$\csc{60^o} = \dfrac{2\sqrt3}{3}$
This means that:
$\csc{(-\frac{13\pi}{3})}
\\=\csc{(-780^o)}
\\=\csc{60^o}
\\= \dfrac{2\sqrt3}{3}$
