Consider the following statements about the graphs of inverse trigonometric functions:
(A) \(y = \sin^{-1} x\) is an increasing function on \([-1, 1]\).
(B) \(y = \cos^{-1} x\) is an increasing function on \([-1, 1]\).
(C) The graph of \(y = \tan^{-1} x\) lies between the lines \(y = -\displaystyle\frac{\pi}{2}\) and \(y = \displaystyle\frac{\pi}{2}\).
(D) \(y = \cot^{-1} x\) is a decreasing function on the set of all real numbers.
Which of the above statements are correct?
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
The graph of an inverse function is the mirror image in the line \(y = x\) of the graph of the function on its principal branch. So it rises where the function rises and falls where the function falls.
\(\sin x\) rises on \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)
\(\cos x\) falls on \([0, \pi]\)
\(\tan x\) rises on \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)
\(\cot x\) falls on \((0, \pi)\)
So \(\sin^{-1}\) and \(\tan^{-1}\) are increasing, \(\cos^{-1}\) and \(\cot^{-1}\) are decreasing, and the values of \(\tan^{-1}\) lie in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).
Checking the statements
(A) \(y = \sin^{-1} x\) is an increasing function on \([-1, 1]\). — Correct. It rises from \(-\frac{\pi}{2}\) at \(x = -1\) to \(\frac{\pi}{2}\) at \(x = 1\).
(B) \(y = \cos^{-1} x\) is an increasing function on \([-1, 1]\). — Incorrect. It falls: \(\cos^{-1}(-1) = \pi\) but \(\cos^{-1}(1) = 0\).
(C) The graph of \(y = \tan^{-1} x\) lies between the lines \(y = -\frac{\pi}{2}\) and \(y = \frac{\pi}{2}\). — Correct. Its values fill \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) without reaching the ends.
(D) \(y = \cot^{-1} x\) is a decreasing function on the set of all real numbers. — Correct. For example \(\cot^{-1}(-1) = \frac{3\pi}{4}\) but \(\cot^{-1}(1) = \frac{\pi}{4}\).