Answer:
To evaluate the expression \(\arcsin\left(\frac{1}{2}\right)\), follow these steps:
1. **Understand the Function:**
- \(\arcsin(x)\) is the inverse function of \(\sin(x)\).
- This means \(\arcsin(x)\) returns the angle \(\theta\) such that \(\sin(\theta) = x\).
2. **Identify the Range:**
- The range of \(\arcsin(x)\) is \([- \frac{\pi}{2}, \frac{\pi}{2}]\). This is the interval in which \(\theta\) will lie.
3. **Solve for the Angle:**
- We need to find the angle \(\theta\) such that \(\sin(\theta) = \frac{1}{2}\).
- Recall the unit circle or trigonometric values: \(\sin(\frac{\pi}{6}) = \frac{1}{2}\).
4. **Verify the Angle Lies in the Range:**
- \(\frac{\pi}{6}\) is within the range \([- \frac{\pi}{2}, \frac{\pi}{2}]\).
5. **Conclusion:**
- Therefore, \(\arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}\).
So, the value of \(\arcsin\left(\frac{1}{2}\right)\) is \(\frac{\pi}{6}\).
