If 4\sin^{-1}x+\cos^{-1}x=\pi, then what is \sin^{-1}x+4\cos^{-1}x equal to?
- A. \pi/2
- B. \pi
- C. 3\pi/2 ✓
- D. 2\pi
Correct Answer: C. 3\pi/2
Explanation
Given 4\sin^{-1}x + \cos^{-1}x = \pi. We know \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}. Substituting this gives 3\sin^{-1}x + \frac{\pi}{2} = \pi, so 3\sin^{-1}x = \frac{\pi}{2} \implies \sin^{-1}x = \frac{\pi}{6}. Then \cos^{-1}x = \frac{\pi}{2} - \frac{\pi}{6} = \frac{\pi}{3}. We need to find \sin^{-1}x + 4\cos^{-1}x = \frac{\pi}{6} + 4(\frac{\pi}{3}) = \frac{\pi}{6} + \frac{8\pi}{6} = \frac{9\pi}{6} = \frac{3\pi}{2}.