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: $(2\pi - \theta)$
First, we make the angle positive:
$-\dfrac{2 \pi}{3}+2\pi=\dfrac{4\pi}{3}$
We are in the 3rd quadrant, so subtract $\pi$:
$\dfrac{4\pi}{3}-\pi=\dfrac{\pi}{3}$
Thus, we have $\csc \dfrac{\pi}{3}=- \dfrac{2 \sqrt 3}{3}$
The value is negative because initially we are in the 3rd qauadrant, where $\csc$ is negative.
