Question

Find the principal value of \[ \tan^{-1}(1) + \cos^{-1} \left(-\frac{1}{2}\right) + \sin^{-1} \left(-\frac{1}{\sqrt{2}}\right). \]

Hint:

Always compute inverse trigonometric values in their principal ranges.

Answer 1

Step 1: Evaluate each term
\[ \tan^{-1}(1) = \frac{\pi}{4}, \quad \cos^{-1}\left(-\frac{1}{2}\right) = \pi - \frac{\pi}{3}, \quad \sin^{-1}\left(-\frac{1}{\sqrt{2}}\right) = -\frac{\pi}{4}. \]
Step 2: Add the terms
\[ \frac{\pi}{4} + \left(\pi - \frac{\pi}{3}\right) - \frac{\pi}{4}. \] Simplify: \[ \frac{\pi}{4} + \frac{2\pi}{3} - \frac{\pi}{4} = \frac{2\pi}{3}. \]
Conclusion: The principal value is \( \frac{2\pi}{3} \).