Question

The value of ${\sin }^{-1}\left(\sin 5\right)-{\cos }^{-1}\left(\cos 5\right)$ is

• A. $0$
• B. $2\pi -10$
• C. $10-4\pi$
• D. $3\pi -10$

Answer 1

The correct option is C $10-4\pi$

$\because {\sin }^{-1}\left(\sin \right(x\left)\right)=x$ for $x\in [-\frac{\pi }{2},\frac{\pi }{2}]$
and
$\because {\cos }^{-1}\left(\cos \right(x\left)\right)=x$ for $x\in [0,\pi ]$
Value of ${\sin }^{-1}\left(\sin 5\right)-{\cos }^{-1}\left(\cos 5\right)$
$={\sin }^{-1}\left(\sin \right(5-2\pi \left)\right)-{\cos }^{-1}\left(\cos \right(2\pi -5\left)\right)$
$=5-2\pi -2\pi +5$
$=10-4\pi$

Alternate Solution

$\because \frac{3\pi }{2}\le 5\le \frac{5\pi }{2}$
From the above graph
$\therefore {\sin }^{-1}\left(\sin x\right)=x-2\pi ,x\in [\frac{3\pi }{2},\frac{5\pi }{2}]$
Hence, ${\sin }^{-1}\left(\sin 5\right)=5-2\pi$


$\because \pi \le 5\le 2\pi$
From the above graph
$\therefore {\cos }^{-1}\left(\cos x\right)=2\pi -x,x\in [\pi ,2\pi ]$
Hence, ${\cos }^{-1}\left(\cos 5\right)=2\pi -5$

Now value of
${\sin }^{-1}\left(\sin 5\right)-{\cos }^{-1}\left(\cos 5\right)$
$=5-2\pi -(2\pi -5)$
$=10-4\pi$