Find the solutions of the equation in the interval [−2𝜋, 2𝜋]. Use a graphing utility to verify your results. (Enter your answers as a comma-separated list.)
csc(x) = 
2

3
3
x = 

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 \(\frac{2}{\sqrt{3}}\).

---

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!

Find the solutions of the equation in the interval [−2π, 2π]. Use a graphing utility to verify your results. (Enter your answers as a comma-separated list.) csc(x) = 2/√3.

Learn how to solve the trigonometric equation csc(x) = 2/√3 within the interval [−2π, 2π]. This solution covers key trigonometric concepts, including reciprocal functions and general solutions for sine. Suitable for high school students in Grades 10-12.

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