Question
Find the principal value of:
\[ \tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2}\right) \]
Solution
Using standard values:
\[ \tan^{-1}(1) = \frac{\pi}{4} \]
And,
\[ \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \]
(since principal range of \( \cos^{-1}x \) is \( [0, \pi] \))
Therefore,
\[ \frac{\pi}{4} + \frac{2\pi}{3} \]
Taking LCM:
\[ = \frac{3\pi + 8\pi}{12} = \frac{11\pi}{12} \]
Final Answer:
\[ \boxed{\frac{11\pi}{12}} \]
Key Concept
Use standard inverse trigonometric values and simplify carefully.