Principal range of sin⁻¹ is:

To find the principal range of $\sin^{-1}(x)$, we must consider the definition of inverse trigonometric functions. 1. **Function Definition**: The function $f(x) = \sin(x)$ is not one-to-one over its entire domain (all real numbers) because it is periodic. For an inverse to exist, the function must be bijective (one-to-one and onto). 2. **Restriction**: We restrict the domain of $\sin(x)$ to the interval $[-\pi/2, \pi/2]$. In this interval, the sine function monotonically increases from $-1$ to $1$, covering all possible values in its range exactly once. 3. **Inversion**: The range of the original function becomes the domain of the inverse function, and the restricted domain of the original function becomes the **range** of the inverse function. 4. **Conclusion**: Therefore, the principal value branch (range) of $y = \sin^{-1}(x)$ is defined as: $$[-\pi/2, \pi/2]$$ This means $-\pi/2 \le y \le \pi/2$.