Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Function | Range of the principal value branch | ||
| (A) | \(\sin^{-1} x\) | (I) | \((0, \pi)\) |
| (B) | \(\cos^{-1} x\) | (II) | \(\left[-\displaystyle\frac{\pi}{2}, \displaystyle\frac{\pi}{2}\right] - \{0\}\) |
| (C) | \(\cot^{-1} x\) | (III) | \([0, \pi]\) |
| (D) | \(\mathrm{cosec}^{-1} x\) | (IV) | \(\left[-\displaystyle\frac{\pi}{2}, \displaystyle\frac{\pi}{2}\right]\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(IV), (B)-(III), (C)-(I), (D)-(II)
The principal value branch is the interval on which the trigonometric function is one-one and takes all its values. An end point is left out where the function is not defined there.
(A) \(\sin^{-1} x\)
This matches (IV) \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\): sine is defined at both ends, \(\sin\left(-\frac{\pi}{2}\right) = -1\) and \(\sin\frac{\pi}{2} = 1\).
(B) \(\cos^{-1} x\)
This matches (III) \([0, \pi]\): cosine falls from \(\cos 0 = 1\) to \(\cos\pi = -1\), with both ends included.
(C) \(\cot^{-1} x\)
This matches (I) \((0, \pi)\): \(\cot 0\) and \(\cot\pi\) are not defined, so both ends are left out.
(D) \(\mathrm{cosec}^{-1} x\)
This matches (II) \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}\): \(\mathrm{cosec}\,0\) is not defined, so 0 is removed.