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 the expression \( \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) \) asks for the angle whose sine is \( \frac{\sqrt{2}}{2} \).
2. Recall that the sine of \( \frac{\pi}{4} \) is \( \frac{\sqrt{2}}{2} \). Therefore, \( \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4} \).
3. Consider the range of the inverse sine function, which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This means the principal value of \( \sin^{-1} \) will be within this interval.
4. Verify that \( \frac{\pi}{4} \) is within the range of \([-\frac{\pi}{2}, \frac{\pi}{2}]\), confirming it as a valid solution.
5. Conclude that the correct angle corresponding to \( \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) \) is \( \frac{\pi}{4} \).