Question
Evaluate:
\[ \cos^{-1}(\cos \tfrac{5\pi}{3}) + \sin^{-1}(\sin \tfrac{5\pi}{3}) \]
Solution
Step 1: Evaluate \( \cos^{-1}(\cos \tfrac{5\pi}{3}) \)
Principal range of \( \cos^{-1}x \) is:
\[ [0, \pi] \]
\[ \frac{5\pi}{3} = 2\pi – \frac{\pi}{3} \Rightarrow \cos \tfrac{5\pi}{3} = \cos \tfrac{\pi}{3} \]
\[ \cos^{-1}(\cos \tfrac{\pi}{3}) = \frac{\pi}{3} \]
Step 2: Evaluate \( \sin^{-1}(\sin \tfrac{5\pi}{3}) \)
Principal range of \( \sin^{-1}x \):
\[ \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
\[ \frac{5\pi}{3} = 2\pi – \frac{\pi}{3} \Rightarrow \sin \tfrac{5\pi}{3} = -\sin \tfrac{\pi}{3} \]
\[ = -\frac{\sqrt{3}}{2} \Rightarrow \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \]
Final Sum:
\[ \frac{\pi}{3} + \left(-\frac{\pi}{3}\right) = 0 \]
Final Answer:
\[ \boxed{0} \]
Key Concept
Always reduce angles and apply principal value ranges carefully.