Question

State $T$ for true and $F$ for false.
(i) The domain of trigonometric functions can be restricted to any one of their branch (not necessarily principal value) in order to obtain their inverse functions.
(ii) The principal value of
$si{n}^{-1}\left[cos\left(si{n}^{-1}\frac{1}{2}\right)\right]$ is $\frac{\pi }{3}$.
(iii) The minimum value of $n$ for which
$ta{n}^{-1}\frac{n}{\pi }>\frac{\pi }{4}$, $n\in N$ is valid is $5$.
(iv) The graph of inverse trigonometric function can be obtained from the graph of their corresponding function by interchanging $x$ and $y$-axes.

	(i)	(ii)	(iii)	(iv)
(a)	F	T	F	T
(b)	T	T	F	T
(c)	T	T	T	T
(d)	T	T	F	F

• A. a
• B. b
• C. c
• D. d

Answer 1

Answer: (B)

(i) True
(ii) True
$\because si{n}^{-1}\left[cos\left(si{n}^{-1}\frac{1}{2}\right)\right]$
$=si{n}^{-1}\left[cos\left\{si{n}^{-1}\left(sin\frac{\pi }{6}\right)\right\}\right]$
$=si{n}^{-1}\left[cos\frac{\pi }{6}\right]$
$=si{n}^{-1}\left(\frac{\sqrt{3}}{2}\right)$
$=si{n}^{-1}\left(sin\frac{\pi }{3}\right)=\frac{\pi }{3}$
(iii) False
Since, $ta{n}^{-1}\frac{n}{\pi }>\frac{\pi }{4}\Rightarrow \frac{n}{\pi }>tan\frac{\pi }{4}$
$\Rightarrow \frac{n}{\pi }>1\Rightarrow n>\pi$
Hence, the minimum value of $n\in N$ is $4$.
(iv) True