Question
Find the principal value of:
\[ \cos^{-1}(\cos \tfrac{2\pi}{3}) + \sin^{-1}(\sin \tfrac{2\pi}{3}) \]
Solution
Principal value ranges:
- \( \cos^{-1}x \in [0, \pi] \)
- \( \sin^{-1}x \in [-\frac{\pi}{2}, \frac{\pi}{2}] \)
First,
\[ \cos^{-1}(\cos \tfrac{2\pi}{3}) = \tfrac{2\pi}{3} \]
(since \( \tfrac{2\pi}{3} \in [0, \pi] \))
Next,
\[ \sin^{-1}(\sin \tfrac{2\pi}{3}) \]
Since \( \tfrac{2\pi}{3} \notin [-\tfrac{\pi}{2}, \tfrac{\pi}{2}] \), use:
\[ \sin^{-1}(\sin x) = \pi – x \quad \text{for } \frac{\pi}{2} < x < \pi \]
\[ = \pi – \frac{2\pi}{3} = \frac{\pi}{3} \]
Therefore,
\[ \frac{2\pi}{3} + \frac{\pi}{3} = \pi \]
Final Answer:
\[ \boxed{\pi} \]
Key Concept
Always apply principal value ranges carefully for inverse trigonometric functions.