Question
Evaluate:
\[ 2\sec^{-1}(2) + \sin^{-1}\left(\frac{1}{2}\right) \]
Solution
We know:
\[ \sec^{-1}(2) = \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]
So,
\[ 2\sec^{-1}(2) = 2 \cdot \frac{\pi}{3} = \frac{2\pi}{3} \]
Also,
\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]
Therefore,
\[ \frac{2\pi}{3} + \frac{\pi}{6} \]
\[ = \frac{4\pi}{6} + \frac{\pi}{6} = \frac{5\pi}{6} \]
Final Answer:
\[ \boxed{\frac{5\pi}{6}} \]
Key Concept
Convert inverse secant into inverse cosine for easy evaluation.