Arrange the principal values of the following functions in ascending order
(A) \(\mathrm{cosec}^{-1}(2)\)
(B) \(\tan^{-1}(-\sqrt{3})\)
(C) \(\tan^{-1}(1)\)
(D) \(\tan^{-1}\left(\cos\displaystyle\frac{3\pi}{2}\right)\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B), (D), (A), (C)
The principal value branches are \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}\) for \(\mathrm{cosec}^{-1}\) and \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) for \(\tan^{-1}\).
\( \displaystyle \text{(A)}\ \ \mathrm{cosec}^{-1}(2) = \frac{\pi}{6},\ \text{since } \mathrm{cosec}\,\frac{\pi}{6} = 2 \)
\( \displaystyle \text{(B)}\ \ \tan^{-1}(-\sqrt{3}) = -\frac{\pi}{3} \)
\( \displaystyle \text{(C)}\ \ \tan^{-1}(1) = \frac{\pi}{4} \)
\( \displaystyle \text{(D)}\ \ \cos\frac{3\pi}{2} = 0,\ \text{so } \tan^{-1}(0) = 0 \)
So
\( \displaystyle -\frac{\pi}{3} \lt 0 \lt \frac{\pi}{6} \lt \frac{\pi}{4} \)
and the ascending order is (B), (D), (A), (C).