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: \[ \frac{\pi}{4} + \pi - \frac{\pi}{3} - \frac{\pi}{4} = \frac{2\pi}{3}. \] Conclusion: The principal value is \( \frac{2\pi}{3} \).