Let's break down the problem you're working on.

The question involves finding the value of $\sin \left( \frac{5\pi}{3} \right)$ using the unit circle.

To find this, we can follow these steps:

1. Locate the angle: The angle $\frac{5\pi}{3}$ is in radians. We know $2\pi$ is a full rotation, so $\frac{5\pi}{3}$ is just shy of a full rotation, falling in the fourth quadrant.

2. Reference angle: The reference angle is the acute angle formed with the x-axis. To find it, subtract $\frac{5\pi}{3}$ from $2\pi$: $2\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3}$ The reference angle is $\frac{\pi}{3}$.

3. Sine in the fourth quadrant: In the fourth quadrant, sine values are negative. The sine of $\frac{\pi}{3}$ is $\frac{\sqrt{3}}{2}$, so: $\sin \left( \frac{5\pi}{3} \right) = -\frac{\sqrt{3}}{2}$

Thus, the answer is $-\frac{\sqrt{3}}{2}$, which corresponds to option A.

Would you like to explore more about the unit circle or other trigonometric functions? Here are a few related questions:

1. How do you find the cosine of an angle using the unit circle?
2. What are the sine and cosine values for angles like $\frac{\pi}{6}$ or $\frac{\pi}{4}$?
3. How can you convert between radians and degrees?
4. What is the significance of reference angles in trigonometry?
5. How does the sign of trigonometric functions change in different quadrants?

Tip: Always remember the signs of trigonometric functions in each quadrant. In the first quadrant, both sine and cosine are positive, in the second quadrant, sine is positive and cosine is negative, in the third quadrant, both are negative, and in the fourth quadrant, sine is negative while cosine is positive.