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:

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

Would you like more details or have any questions?

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.