Question

If $x,y\in \{1,2,3,4\}$ then the probability that ${\sin }^{-1}(\sin x)+{\cos }^{-1}(\cos y)$ is an integer, is

• A. $\frac{1}{16}$
• B. $\frac{3}{16}$
• C. $\frac{5}{16}$
• D. $\frac{14}{16}$

Answer 1

Answer: (B)

${\sin }^{-1}(\sin 1)=1{\cos }^{-1}(\cos 1)=1$
${\sin }^{-1}(\sin 2)=\pi -2{\cos }^{-1}(\cos 2)=2$
${\sin }^{-1}(\sin 3)=\pi -3{\cos }^{-1}(\cos 3)=3$
${\sin }^{-1}(\sin 4)=\pi -4{\cos }^{-1}(\cos 4)=2\pi -4$
${\sin }^{-1}(\sin x)+{\cos }^{-1}(\cos x)\text{ is an integer }$
$\text{ So, }(x,y)\in (1,1)(1,2)(1,3)$
$\text{ So probability }=\frac{3}{16}$