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} $