Question

Find the value of \[ \sin^{-1} \left( \sin \left( \frac{5\pi}{6} \right) \right). \] The answer is:

• A. \( \frac{5\pi}{6} \)
• B. \( \frac{\pi}{2} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{12} \)
• E. \( \frac{\pi}{6} \)

Hint:

For inverse trigonometric functions, always ensure the result is within the range of the function. If the argument is outside the range, find the corresponding equivalent angle within the domain.

Answer 1

Answer: (E)

We are asked to find the value of \( \sin^{-1}\left(\sin \left(\frac{5\pi}{6}\right)\right) \). First, recall that the function \( \sin^{-1}(x) \) (inverse sine) is defined on the principal range \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \). This means that for any angle \( \theta \), \( \sin^{-1}(\sin \theta) \) will return the value of \( \theta \) within this range.
Now, consider the angle \( \frac{5\pi}{6} \). This angle is greater than \( \frac{\pi}{2} \), so we need to adjust it to fall within the principal range. 
Since \( \sin \left( \frac{5\pi}{6} \right) = \sin \left( \frac{\pi}{6} \right) \), we have:
\[ \sin^{-1} \left( \sin \left( \frac{5\pi}{6} \right) \right) = \sin^{-1} \left( \sin \left( \frac{\pi}{6} \right) \right) \] Therefore, the result is \( \frac{\pi}{6} \), as \( \frac{\pi}{6} \) is within the principal range of \( \sin^{-1} \).
Thus, the correct answer is option (E), \( \frac{\pi}{6} \).