The problem shown in your image is asking for the values of $x$ within a specific interval that satisfy the equation:

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

You are instructed to find the values of $x$ on the interval:

$\left[ -\frac{8\pi}{3}, \frac{3\pi}{2} \right]$

Step-by-Step Approach:

1. Identify the general solutions for $\sin(x) = -\frac{\sqrt{3}}{2}$:

   • The sine function equals $-\frac{\sqrt{3}}{2}$ at the angles $x = \frac{4\pi}{3} + 2n\pi$ and $x = \frac{5\pi}{3} + 2n\pi$, where $n$ is any integer.

2. Restrict the solutions to the interval $\left[ -\frac{8\pi}{3}, \frac{3\pi}{2} \right]$:

   • We need to find all $x$ values in this interval where $\sin(x) = -\frac{\sqrt{3}}{2}$.

3. Check for solutions in the interval $\left[ -\frac{8\pi}{3}, \frac{3\pi}{2} \right]$:

   • For $n = -2, -1, 0, 1$, substitute $n$ into the general solutions to check which values fall within the interval.

Let me calculate the exact values for you.

The values of $x$ that satisfy $\sin(x) = -\frac{\sqrt{3}}{2}$ within the interval $\left[ -\frac{8\pi}{3}, \frac{3\pi}{2} \right]$ are:

$x = -\frac{8\pi}{3}, -\frac{7\pi}{3}, -\frac{2\pi}{3}, -\frac{\pi}{3}, \frac{4\pi}{3}$

These are the solutions you can use in your answer.

Would you like more details on any of the steps, or do you have any further questions?

Here are five related questions to deepen your understanding:

1. What is the general form of solutions for trigonometric equations?
2. How does the sine function behave over different quadrants in the unit circle?
3. Why are there two general solutions for $\sin(x) = -\frac{\sqrt{3}}{2}$?
4. How do you determine which solutions fall within a specific interval?
5. Can you find solutions to similar problems with other trigonometric functions like $\cos(x)$ or $\tan(x)$?

Tip: Always visualize the unit circle to better understand where trigonometric values occur across different quadrants.