Question

The value of ${sin}^{-1}\left(sin12\right)+{cos}^{-1}\left(cos12\right)$ is equal to :

• A. $Zero$
• B. $24-2\pi$
• C. $4\pi -24$
• D. None of these

Answer 1

The correct option is A $Zero$

Given,

${\sin }^{-1}\left(\sin 12\right)+{\cos }^{-1}\left(\cos 12\right)$

The principle values,

${\sin }^{-1}x\in [-\frac{\pi }{2},\frac{\pi }{2}],\forall x\in [-1,1]$

${\cos }^{-1}x\in [0,\pi ],\forall x\in [-1,1]$

$\therefore {\sin }^{-1}\left(\sin 12\right)\ne 12\notin [-\frac{\pi }{2},\frac{\pi }{2}]$

$\therefore {\cos }^{-1}\left(\cos 12\right)\ne 12\notin [0,\pi ]$

$\therefore {\sin }^{-1}\left(\sin 12\right)+{\cos }^{-1}\left(\cos 12\right)$

$={\sin }^{-1}\left(\sin \right(12-4\pi \left)\right)+{\cos }^{-1}\left(\cos \right(4\pi -12\left)\right)$

$=(12-4\pi )+(4\pi -12)$

$=0$

$\therefore {\sin }^{-1}\left(\sin 12\right)+{\cos }^{-1}\left(\cos 12\right)=0$