What is the range of $y={\sin }^{-1}x$?

• A. $(-\infty ,\infty )$
• B. $[-\pi ,\pi ]$
• C. $[-\frac{\pi }{2},\frac{\pi }{2}]$
• D. $[0,\pi ]$

Answer: (C)

Verified step by step guidance

1. Recall that the function \( y = \sin^{-1} x \) (also called arcsine) is the inverse of the sine function restricted to its principal domain. This means it takes an input \( x \) in the interval \( [-1, 1] \) and returns an angle \( y \) whose sine is \( x \).
2. Understand that the domain of \( y = \sin^{-1} x \) is \( [-1, 1] \), because sine values only range between -1 and 1. The question asks for the range of \( y \), which is the set of possible output values (angles).
3. The principal range of the arcsine function is the interval where the sine function is one-to-one and covers all values from -1 to 1. This interval is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), meaning the output angles are between \( -90^\circ \) and \( 90^\circ \) in radians.
4. Therefore, the range of \( y = \sin^{-1} x \) is the set of all angles \( y \) such that \( y \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \).
5. Summarize: The range of the arcsine function is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), which corresponds to the interval of output values for \( y = \sin^{-1} x \).