Find the principal value of sin inverse (-1) input in radians

The principal value of \( \sin^{-1}(-1) \) (also written as \( \arcsin(-1) \)) is the angle \( \theta \) such that \( \sin(\theta) = -1 \) and \( \theta \) is within the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) (the principal range for arcsine).

Since \( \sin(-\frac{\pi}{2}) = -1 \), the principal value of \( \sin^{-1}(-1) \) is:

\[
\sin^{-1}(-1) = -\frac{\pi}{2} \, \text{radians}
\]

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Here are 5 related questions to deepen your understanding:
1. What is the principal value of \( \sin^{-1}(0) \)?
2. How is the range of \( \sin^{-1}(x) \) different from \( \cos^{-1}(x) \)?
3. What is the general solution of \( \sin(\theta) = -1 \) in radians?
4. How do you compute \( \sin^{-1}(x) \) for values outside \( [-1, 1] \)?
5. What is the difference between \( \sin^{-1}(x) \) and \( \csc^{-1}(x) \)?

**Tip:** For inverse trigonometric functions, always check the range of principal values.

Discover how to find the principal value of sin^{-1}(-1) in radians. This problem involves key trigonometric concepts such as inverse sine, and the principal range for arcsine. Suitable for students in Grades 10-12.

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