Match List-I with List-II
| List-I | List-II |
| Inverse Trigonometric function | Principal values |
| (A) \( \sec^{-1}(-2) \) | (I) \( \displaystyle\frac{5\pi}{6} \) |
| (B) \( \mathrm{cosec}^{-1}(-\sqrt{2}) \) | (II) \( \displaystyle\frac{2\pi}{3} \) |
| (C) \( \mathrm{cosec}^{-1}(2) \) | (III) \( -\displaystyle\frac{\pi}{4} \) |
| (D) \( \sec^{-1}\left(-\displaystyle\frac{2}{\sqrt{3}}\right) \) | (IV) \( \displaystyle\frac{\pi}{6} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) - (II), (B) - (III), (C) - (IV), (D) - (I)
The principal range of \( \sec^{-1} \) is \( [0,\pi]-\left\{\frac{\pi}{2}\right\} \), and that of \( \mathrm{cosec}^{-1} \) is \( \left[-\frac{\pi}{2},\frac{\pi}{2}\right]-\{0\} \).
(A) \( \sec^{-1}(-2) \)
This matches (II) \( \frac{2\pi}{3} \), since \( \sec\frac{2\pi}{3}=-2 \).
(B) \( \mathrm{cosec}^{-1}(-\sqrt{2}) \)
This matches (III) \( -\frac{\pi}{4} \), since \( \mathrm{cosec}\left(-\frac{\pi}{4}\right)=-\sqrt{2} \).
(C) \( \mathrm{cosec}^{-1}(2) \)
This matches (IV) \( \frac{\pi}{6} \), since \( \mathrm{cosec}\frac{\pi}{6}=2 \).
(D) \( \sec^{-1}\left(-\frac{2}{\sqrt{3}}\right) \)
This matches (I) \( \frac{5\pi}{6} \), since \( \sec\frac{5\pi}{6}=-\frac{2}{\sqrt{3}} \).
So the matching is (A) - (II), (B) - (III), (C) - (IV), (D) - (I).