Question
Find the value of:
\[ \sin\left(\frac{\pi}{3} – \sin^{-1}\left(-\frac{1}{2}\right)\right) \]
Solution
First, evaluate:
\[ \sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6} \]
(since principal range of \( \sin^{-1}x \) is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \))
Now substitute:
\[ \sin\left(\frac{\pi}{3} – (-\frac{\pi}{6})\right) = \sin\left(\frac{\pi}{3} + \frac{\pi}{6}\right) \]
\[ = \sin\left(\frac{\pi}{2}\right) \]
\[ = 1 \]
Final Answer:
\[ \boxed{1} \]
Key Concept
Evaluate inverse trig first, then simplify using standard angle values.