Question

If $si{n}^{-1}\left(\frac{5}{x}\right)+si{n}^{-1}\left(\frac{12}{x}\right)=\frac{\pi }{2}$, then $x$ is equal to _____.

Answer 1

Answer: 13

Put $si{n}^{-1}\frac{5}{x}=A$ or $\frac{5}{x}=sinA$
$si{n}^{-1}\frac{12}{x}=B$ or $\frac{12}{x}=sinB$
$\Rightarrow A+B=\frac{\pi }{2}$
$sinA=sin(\frac{\pi }{2}-B)=cosB=\sqrt{1-si{n}^{2}B}$
$\Rightarrow \frac{5}{x}=\sqrt{1-\frac{144}{x^2}}$ or $\frac{169}{x^2}=1$
or $x^2=169$ or $x=13$
[$\because x=-13$ does not satisfy the given equation]