To find the exact value of $\sin\left( \frac{\pi}{3} \right)$, we can use the standard values from the unit circle.

Step-by-step:

The angle $\frac{\pi}{3}$ radians corresponds to $60^\circ$ in degrees.

From the unit circle, the sine of $60^\circ$ (or $\frac{\pi}{3}$ radians) is:

$\sin\left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2}$

Conclusion:

The exact value of $\sin\left( \frac{\pi}{3} \right)$ is $\frac{\sqrt{3}}{2}$.

Let me know if you need any further explanation or if you have any questions!

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Follow-up questions:

1. How do we determine sine values from the unit circle?
2. How do radians and degrees relate to each other?
3. What are some other key sine values on the unit circle?
4. How would you calculate the cosine of $\frac{\pi}{3}$?
5. How does the unit circle relate to trigonometric identities?

Tip:

To memorize key values like $\sin\left( \frac{\pi}{3} \right)$, it’s helpful to remember the sine values for $30^\circ$, $45^\circ$, and $60^\circ$ as they often come up in problems.