Question

Consider the following statement :
(i) $si{n}^{-1}\left(sin\frac{2\pi }{3}\right)=\frac{2\pi }{3}$
(ii) $ta{n}^{-1}\left(tan\frac{2\pi }{3}\right)=\frac{-\pi }{3}$
(iii) $co{s}^{-1}\left(cos\frac{2\pi }{3}\right)=\frac{2\pi }{3}$
(iv) $se{c}^{-1}\left(sec\frac{5\pi }{4}\right)=\frac{3\pi }{4}$

• A. (i), (ii), (iii)
• B. (ii), (iii), (iv)
• C. (i), (ii), (iv)
• D. None of these

Answer 1

The correct option is B (ii), (iii), (iv)

$si{n}^{-1}\left(sin\frac{2\pi }{3}\right)=si{n}^{-1}\left(sin\left(\frac{\pi -\pi }{3}\right)\right)$
$=si{n}^{-1}\left(sin\frac{\pi }{3}\right)=\frac{\pi }{3}$
$\therefore$ (i) is wrong.
$ta{n}^{-1}\left(sin\frac{2\pi }{3}\right)=ta{n}^{-1}tan(-\pi +\frac{2\pi }{3})$
$=ta{n}^{-1}tan\left(\frac{-\pi }{3}\right)=\frac{-\pi }{3}$
$\therefore$ (ii) is true.
$co{s}^{-1}\left(cos\frac{2\pi }{3}\right)=\frac{2\pi }{3}as\frac{2\pi }{3}ϵ[0,2\pi ]$
$\therefore$ (iii) is true
$se{c}^{-1}\left(sec\frac{5\pi }{4}\right)=se{c}^{-1}sec(2\pi -\frac{3\pi }{4})$
$=se{c}^{-1}sec\frac{3\pi }{4}=\frac{3\pi }{4}$
$\therefore$ (iv) is true
Hence (ii), (iii) and (iv) are true.