Evaluate the expression.
$\sin^{-1}\left(\frac{\sqrt2}{2}\right)$

• A. $\frac{\pi}{4}$
• B. $-\frac{\pi}{4}$
• C. $\frac{3\pi}{4}$
• D. $\frac{7\pi}{4}$

Answer: (A)

Verified step by step guidance

1. Understand that \( \sin^{-1} \) is the inverse sine function, also known as arcsin, which gives the angle whose sine is a given number.
2. Recognize that \( \frac{\sqrt{2}}{2} \) is a well-known sine value corresponding to specific angles on the unit circle.
3. Recall that \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \) and \( \sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2} \). These angles are in the first and second quadrants, respectively.
4. Determine the principal value range for \( \sin^{-1} \), which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). Within this range, the angle that corresponds to \( \frac{\sqrt{2}}{2} \) is \( \frac{\pi}{4} \).
5. Conclude that \( \sin^{-1}(\frac{\sqrt{2}}{2}) = \frac{\pi}{4} \) because it falls within the principal value range of the inverse sine function.