Consider the following expressions:
(A) \(\sin^{-1}\left(\displaystyle\frac{\pi}{4}\right)\)
(B) \(\cos^{-1}\left(\sqrt{3}\right)\)
(C) \(\sec^{-1}\left(\displaystyle\frac{1}{2}\right)\)
(D) \(\cot^{-1}(0)\)
Which of the above are defined as real numbers?
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (D) only
An expression is defined only when the number inside lies in the domain of the inverse function:
\(\sin^{-1}\) and \(\cos^{-1}\): \([-1, 1]\)
\(\sec^{-1}\): \(\mathrm{R} - (-1, 1)\)
\(\cot^{-1}\): all real numbers
The numbers inside are
\( \displaystyle \frac{\pi}{4} \approx 0.785, \quad \sqrt{3} \approx 1.732, \quad \frac{1}{2} = 0.5, \quad 0 \)
So \(\frac{\pi}{4}\) is in \([-1, 1]\), \(\sqrt{3}\) is not, \(\frac{1}{2}\) is in \((-1, 1)\), and 0 is a real number.
Checking the statements
(A) \(\sin^{-1}\left(\frac{\pi}{4}\right)\) — Correct. \(\frac{\pi}{4} \approx 0.785\) lies in \([-1, 1]\).
(B) \(\cos^{-1}\left(\sqrt{3}\right)\) — Incorrect. \(\sqrt{3} \approx 1.732\) is greater than 1.
(C) \(\sec^{-1}\left(\frac{1}{2}\right)\) — Incorrect. \(\frac{1}{2}\) lies in \((-1, 1)\), and no secant has a value there.
(D) \(\cot^{-1}(0)\) — Correct. It is defined and equals \(\frac{\pi}{2}\).