To solve the equation $\csc(x) = \frac{2\sqrt{3}}{3}$ in the interval $[-2\pi, 2\pi]$, follow these steps:

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Step 1: Express the equation in terms of $\sin(x)$

The cosecant function is the reciprocal of the sine function: $\csc(x) = \frac{1}{\sin(x)}.$ This gives: $\frac{1}{\sin(x)} = \frac{2\sqrt{3}}{3}.$ Rearranging: $\sin(x) = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2}.$

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Step 2: Determine the general solutions

The sine function $\sin(x) = \frac{\sqrt{3}}{2}$ corresponds to angles where sine equals $\frac{\sqrt{3}}{2}$. In one period ($[0, 2\pi]$), these angles are: $x = \frac{\pi}{3}, \frac{2\pi}{3}.$

Step 3: Extend the solutions to the interval $[-2\pi, 2\pi]$

The sine function is periodic with period $2\pi$. To find all solutions in the given interval:

1. For $\frac{\pi}{3}$: $x = \frac{\pi}{3}, \frac{\pi}{3} - 2\pi = -\frac{5\pi}{3}, \frac{\pi}{3} + 2\pi = \text{(outside the interval)}.$
2. For $\frac{2\pi}{3}$: $x = \frac{2\pi}{3}, \frac{2\pi}{3} - 2\pi = -\frac{4\pi}{3}, \frac{2\pi}{3} + 2\pi = \text{(outside the interval)}.$

Thus, the solutions are: $x = \pm \frac{\pi}{3}, \pm \frac{2\pi}{3}.$

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Step 4: Final solution

The solutions in the interval $[-2\pi, 2\pi]$ are: $x = -\frac{5\pi}{3}, -\frac{4\pi}{3}, \frac{\pi}{3}, \frac{2\pi}{3}.$

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Step 5: Verify using a graphing utility

Graph $y = \csc(x)$ and check where it equals $\frac{2\sqrt{3}}{3}$. Confirm that the solutions match the computed values.

Would you like me to provide a graph or additional explanation?

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5 Related Questions:

1. What is the period of the $\csc(x)$ function?
2. How do you determine the general solutions for $\sin(x) = k$?
3. What are the key properties of $\csc(x)$?
4. How does the reciprocal relationship between $\sin(x)$ and $\csc(x)$ affect their graphs?
5. Can $\csc(x)$ ever be undefined?

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Tip:

For trigonometric equations involving reciprocal functions, it can be helpful to first rewrite them in terms of sine or cosine to simplify calculations.