Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Equation | Value of x | ||
| (A) | \(\sin^{-1} x = -\displaystyle\frac{\pi}{4}\) | (I) | \(-\displaystyle\frac{\sqrt{3}}{2}\) |
| (B) | \(\cos^{-1} x = \displaystyle\frac{5\pi}{6}\) | (II) | \(-\displaystyle\frac{2}{\sqrt{3}}\) |
| (C) | \(\tan^{-1} x = -\displaystyle\frac{\pi}{3}\) | (III) | \(-\displaystyle\frac{1}{\sqrt{2}}\) |
| (D) | \(\mathrm{cosec}^{-1} x = -\displaystyle\frac{\pi}{3}\) | (IV) | \(-\sqrt{3}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(III), (B)-(I), (C)-(IV), (D)-(II)
Each given angle lies in the principal value branch of its function, so by the definition of the inverse function, x is the trigonometric value of that angle.
(A) \(\sin^{-1} x = -\frac{\pi}{4}\)
This matches (III) \(-\frac{1}{\sqrt{2}}\): \(x = \sin\left(-\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}}\).
(B) \(\cos^{-1} x = \frac{5\pi}{6}\)
This matches (I) \(-\frac{\sqrt{3}}{2}\): \(x = \cos\frac{5\pi}{6} = -\cos\frac{\pi}{6} = -\frac{\sqrt{3}}{2}\).
(C) \(\tan^{-1} x = -\frac{\pi}{3}\)
This matches (IV) \(-\sqrt{3}\): \(x = \tan\left(-\frac{\pi}{3}\right) = -\sqrt{3}\).
(D) \(\mathrm{cosec}^{-1} x = -\frac{\pi}{3}\)
This matches (II) \(-\frac{2}{\sqrt{3}}\): \(\sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\), so \(x = \mathrm{cosec}\left(-\frac{\pi}{3}\right) = -\frac{2}{\sqrt{3}}\).