The value of $\arcsin\left(\frac{1}{\sqrt{3}}\right)$ represents the angle $\theta$ such that $\sin(\theta) = \frac{1}{\sqrt{3}}$. Let's solve this step by step:

Step 1: Understanding the range of $\arcsin$

The $\arcsin(x)$ function gives the angle in the range $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$.

Step 2: Solve $\sin(\theta) = \frac{1}{\sqrt{3}}$

We know $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. If $\sin(\theta) = \frac{1}{\sqrt{3}}$, this corresponds to the angle where such a ratio occurs.

The exact value of this angle is $\arcsin\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \approx 0.5236 \, \text{radians}$.

Final Answer:

$\arcsin\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}$

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Would you like further clarification or a detailed derivation? Here are some questions to consider:

1. What are the properties of the $\arcsin$ function?
2. How does the $\sin$ function relate to the unit circle?
3. Can you express $\frac{1}{\sqrt{3}}$ in terms of trigonometric values of common angles?
4. How does the range of $\arcsin(x)$ affect solutions?
5. What are the equivalent angles in degrees for $\frac{\pi}{6}$?

Tip: Memorizing the sine values of common angles like $30^\circ, 45^\circ,$ and $60^\circ$ helps with quick calculations!