Evaluate: cos-1(1/2) − 2sin-1(−1/2)
Solution:
We know the principal values:
\[ \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]
\[ \sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6} \]
(Since the range of sin-1(x) is \([- \pi/2, \pi/2]\))
Substitute values:
\[ \cos^{-1}\left(\frac{1}{2}\right) – 2\sin^{-1}\left(-\frac{1}{2}\right) = \frac{\pi}{3} – 2\left(-\frac{\pi}{6}\right) \]
\[ = \frac{\pi}{3} + \frac{\pi}{3} = \frac{2\pi}{3} \]
Final Answer:
Value = \[ \frac{2\pi}{3} \]