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What is the range of $\arcsin x$ ?

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Polynomial vocabulary

Full solution

Q. What is the range of

\arcsin x

?

Definition of arcsin function: The arcsin function, also known as the inverse sine function, is defined only for inputs between $-1$ and $1$ , inclusive. This is because the sine function, which arcsin is the inverse of, has outputs that range from $-1$ to $1$ .
Range of sine function: Since the $\text{arcsin}$ function is the inverse of the sine function, we need to consider the range of angles for which the sine function gives all possible values between $-1$ and $1$ . The sine function achieves all values between $-1$ and $1$ in the interval from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ radians.
Range of arcsin function: Therefore, the range of the arcsin function is the set of all real numbers from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ , including these endpoints. This is because for every value $y$ in the interval $[-1, 1]$ , there is a corresponding angle $x$ in the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$ such that $\sin(x) = y$ .

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