Question

Find the principal value of:
\( \cos^{-1}\left(-\frac{1}{2}\right) + 2\sin^{-1}(1) \)

Answer 1

Step 1: Evaluate \( \cos^{-1}\left(-\frac{1}{2}\right) \)
Using identity: \[ \cos^{-1}(-x) = \pi - \cos^{-1}(x) \] So, \[ \cos^{-1}\left(-\frac{1}{2}\right) = \pi - \cos^{-1}\left(\frac{1}{2}\right) \] Now, \[ \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \Rightarrow \cos^{-1}\left(-\frac{1}{2}\right) = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

Step 2: Evaluate \( \sin^{-1}\left(\frac{1}{2}\right) \)
From standard value: \[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]

Step 3: Multiply and Add
\[ \cos^{-1}\left(-\frac{1}{2}\right) + 2\sin^{-1}\left(\frac{1}{2}\right) = \frac{2\pi}{3} + 2 \cdot \frac{\pi}{6} = \frac{2\pi}{3} + \frac{\pi}{3} = \pi \]

Final Answer:

\[ \boxed{\pi} \]