Question

The probability that ${\sin }^{-1}\left(\sin x\right)+{\cos }^{-1}\left(\cos y\right)$, where $x,y\in \{1,2,3,4\}$, is an integer is

• A. $\frac{1}{16}$
• B. $\frac{3}{16}$
• C. $\frac{15}{16}$
• D. $\frac{1}{2}$

Answer 1

The correct option is B $\frac{3}{16}$

For ${\sin }^{-1}\left(\sin x\right)+{\cos }^{-1}\left(\cos y\right)$ to be an integer, integral $x$ should lie between $[-\frac{\pi }{2},\frac{\pi }{2}]$ and $y\in [0,\pi ]$.
$\Rightarrow x=1$ and $y=1,2,3$.
$\therefore$ Required probability $y=\frac{3}{16}$