Evaluate the expression.
$\sin^{-1}\left(\frac{\sqrt3}{2}\right)$

• A. $\frac{\pi}{6}$
• B. $\frac{3\pi}{2}$
• C. $\frac{\pi}{3}$
• D. $-\frac{\pi}{3}$

Answer: (C)

Verified step by step guidance

1. Understand that the problem involves evaluating the inverse sine function, \( \sin^{-1} \), which gives the angle whose sine is the given value.
2. Recognize that \( \sin^{-1} \left( \frac{\sqrt{3}}{2} \right) \) asks for the angle \( \theta \) such that \( \sin(\theta) = \frac{\sqrt{3}}{2} \).
3. Recall that the sine of \( \frac{\pi}{3} \) is \( \frac{\sqrt{3}}{2} \), which is a common angle in the unit circle.
4. Consider the range of the inverse sine function, which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), to determine the principal value of the angle.
5. Conclude that the principal value of \( \sin^{-1} \left( \frac{\sqrt{3}}{2} \right) \) is \( \frac{\pi}{3} \), as it falls within the range of the inverse sine function.