We are given the equation:

$\csc(x) = \frac{2}{\sqrt{3}}$

First, let's express the equation in terms of sine since the cosecant function is the reciprocal of the sine function:

$\csc(x) = \frac{1}{\sin(x)} \implies \sin(x) = \frac{\sqrt{3}}{2}$

Next, we need to find the values of $x$ for which $\sin(x) = \frac{\sqrt{3}}{2}$ within the interval $[-2\pi, 2\pi]$. The general solutions for $\sin(x) = \frac{\sqrt{3}}{2}$ occur at:

$x = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad x = \frac{2\pi}{3} + 2n\pi$ where $n$ is any integer.

Now, let's find the specific solutions within the interval $[-2\pi, 2\pi]$:

1. Starting with $x = \frac{\pi}{3}$, we can add or subtract multiples of $2\pi$ to find other solutions:

   • $x = \frac{\pi}{3}$
   • $x = \frac{\pi}{3} - 2\pi = \frac{\pi}{3} - \frac{6\pi}{3} = -\frac{5\pi}{3}$

2. For $x = \frac{2\pi}{3}$, similarly:

   • $x = \frac{2\pi}{3}$
   • $x = \frac{2\pi}{3} - 2\pi = \frac{2\pi}{3} - \frac{6\pi}{3} = -\frac{4\pi}{3}$

Thus, the solutions within $[-2\pi, 2\pi]$ are:

$x = -\frac{5\pi}{3}, -\frac{4\pi}{3}, \frac{\pi}{3}, \frac{2\pi}{3}$

To verify the results using a graphing utility, plot $y = \csc(x)$ and check where it equals $\frac{2}{\sqrt{3}}$.

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Would you like more details on verifying with a graphing tool or any other questions? Here are some related questions:

1. How can you solve equations involving other trigonometric functions in a given interval?
2. What are the general solutions for $\sin(x) = \frac{1}{2}$?
3. How do transformations affect the graph of a sine or cosine function?
4. What is the relationship between the sine and cosecant functions?
5. How can you solve trigonometric equations using identities?

Tip: When solving trigonometric equations, always express the equation in terms of sine or cosine if possible for easier handling!