Question
Find the value of:
\[ 2\sin^{-1}\left(\frac{1}{2}\right) + \cos^{-1}\left(-\frac{1}{2}\right) \]
Solution
Using standard values:
\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]
So,
\[ 2\sin^{-1}\left(\frac{1}{2}\right) = 2 \cdot \frac{\pi}{6} = \frac{\pi}{3} \]
Also,
\[ \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \]
(since principal range of \( \cos^{-1}x \) is \( [0, \pi] \))
Therefore,
\[ \frac{\pi}{3} + \frac{2\pi}{3} = \pi \]
Final Answer:
\[ \boxed{\pi} \]
Key Concept
Use standard inverse trigonometric values within principal ranges.