Question

Assertion (A) : $sin^{-1} (sin \,5) = 5$ (where $5$ is in radians)
Reason (R) : $sin^{-1}(sin\, \theta) = 0$ for principal value.

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true but (R) is false.
• D. (A) is false but (R) is true.

Answer 1

Answer: (D)

$sin\,5 = sin(\pi + \overline{5-\pi})$
$ = -sin(5 - \pi), 5 - \pi \notin [-\frac{\pi}{2}, \frac{\pi}{2}] = -sin(\pi + \overline{5 - 2\pi})$
$= sin (5 - 2\pi)$, where $5 - 2\pi \in[\frac{\pi}{2}, \frac{\pi}{2}]$
$\therefore sin^{-1} (sin\,5) = sin^{-1} sin(5 - 2\pi)$
$\Rightarrow sin^{-1} (sin \,5) = 5 - 2 \,\pi \ne 5$
$\therefore $ Assertion (A) is false but given
Reason (R) is true.