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The problem is asking for the exact value of \( \sin^{-1} \left( \sin \left( \frac{5\pi}{6} \right) \right) \).

### Step-by-Step Solution:

1. **Understanding the inverse sine (arcsine)**:
   - The function \( \sin^{-1}(x) \) (also called arcsine) returns the angle \( y \) such that \( \sin(y) = x \).
   - The principal range of \( \sin^{-1}(x) \) is \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \). This is crucial because arcsine only returns values in this range.

2. **Simplifying \( \sin \left( \frac{5\pi}{6} \right) \)**:
   - \( \frac{5\pi}{6} \) is in the second quadrant where sine is positive.
   - The reference angle for \( \frac{5\pi}{6} \) is \( \frac{\pi}{6} \), so \( \sin \left( \frac{5\pi}{6} \right) = \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \).

3. **Finding \( \sin^{-1} \left( \frac{1}{2} \right) \)**:
   - We now need to find the angle \( \theta \) such that \( \sin(\theta) = \frac{1}{2} \) and \( \theta \) lies in the range \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \).
   - The angle in this range is \( \frac{\pi}{6} \).

### Conclusion:
The exact value of \( \sin^{-1} \left( \sin \left( \frac{5\pi}{6} \right) \right) \) is \( \frac{\pi}{6} \).

So, the correct answer is:
\[
\boxed{A} \quad \frac{\pi}{6}
\]

Would you like more details on inverse trigonometric functions?

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Here are 5 related questions for further practice:

1. What is the value of \( \sin^{-1} \left( \sin \left( \frac{7\pi}{6} \right) \right) \)?
2. Simplify \( \sin^{-1} \left( \sin \left( -\frac{3\pi}{4} \right) \right) \).
3. What is the principal value of \( \cos^{-1} \left( \cos \left( \frac{4\pi}{3} \right) \right) \)?
4. Find \( \tan^{-1} \left( \tan \left( \frac{5\pi}{4} \right) \right) \).
5. What is the range of the inverse sine function?

**Tip:** When solving problems involving inverse trigonometric functions, always pay attention to the principal range of the inverse function.

Find the exact value of sin^(-1)(sin(5π/6)).

Find the exact value of sin^(-1)(sin(5π/6)) with a detailed step-by-step explanation. Learn about inverse trigonometric functions, their principal ranges, and reference angles. Suitable for students in Grades 11-12.

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