Answer
$\dfrac{-2\sqrt 3}{3}$
Work Step by Step
The reference angle of an angle $0 \leq \theta \lt 2\pi $ based on its position can be computed by using the following steps:
a) Quadrant- I: $\theta $
b) Quadrant -II: $(\pi-\theta)$
c) Quadrant- III: $(\theta - \pi)$
d) Quadrant - IV: $(2\pi - \theta)$
We can see that the angle $\dfrac{22\pi}{3}$ is close to $\dfrac{21\pi}{3}=7\pi $.
Thus, the reference angle of $\dfrac{22\pi}{3}$ is:
$ \dfrac{22\pi}{3}-7 \pi =\dfrac{22\pi}{3} - \dfrac{21\pi}{3}=\dfrac{\pi}{3}$
$\implies \csc \dfrac{\pi}{3} =\dfrac{2\sqrt 3}{3}$
So, $\csc (\dfrac{22\pi}{3})=\dfrac{-2\sqrt 3}{3}$ because $\theta $ lies in Quadrant-III.
