Consider the following statements:
(A) \(\cos^{-1}(0) = \displaystyle\frac{\pi}{2}\)
(B) \(\sin^{-1}(-1) = \displaystyle\frac{3\pi}{2}\)
(C) \(\sec^{-1}(-1) = \pi\)
(D) \(\tan^{-1}(0) = 0\)
Which of the above statements are correct?
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only
Each value must be the angle in the principal value branch with the given trigonometric value: \([0, \pi]\) for \(\cos^{-1}\), \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) for \(\sin^{-1}\), \([0, \pi] - \left\{\frac{\pi}{2}\right\}\) for \(\sec^{-1}\) and \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) for \(\tan^{-1}\).
\( \displaystyle \cos\frac{\pi}{2} = 0, \quad \sin\left(-\frac{\pi}{2}\right) = -1, \quad \sec\pi = \frac{1}{\cos\pi} = -1, \quad \tan 0 = 0 \)
So \(\cos^{-1}(0) = \frac{\pi}{2}\), \(\sin^{-1}(-1) = -\frac{\pi}{2}\), \(\sec^{-1}(-1) = \pi\) and \(\tan^{-1}(0) = 0\).
Checking the statements
(A) \(\cos^{-1}(0) = \frac{\pi}{2}\) — Correct. \(\cos\frac{\pi}{2} = 0\) and \(\frac{\pi}{2}\) is in \([0, \pi]\).
(B) \(\sin^{-1}(-1) = \frac{3\pi}{2}\) — Incorrect. \(\sin\frac{3\pi}{2} = -1\), but \(\frac{3\pi}{2}\) is outside \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\); the value is \(-\frac{\pi}{2}\).
(C) \(\sec^{-1}(-1) = \pi\) — Correct. \(\sec\pi = -1\) and \(\pi\) is the right end of the branch \([0, \pi] - \left\{\frac{\pi}{2}\right\}\).
(D) \(\tan^{-1}(0) = 0\) — Correct. \(\tan 0 = 0\) and 0 is in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).