Question

Solve: ${\sin }^{-1}\frac{5}{x}+{\sin }^{-1}\frac{12}{x}=\frac{\pi }{2}$

Answer 1

${\sin }^{-1}\frac{5}{x}+{\sin }^{-1}\frac{12}{x}=\frac{\pi }{2}$

Let $\beta ={\sin }^{-1}\frac{12}{x}$

$\Rightarrow \sin \beta =\frac{12}{x}$

We know that ${\cos }^2\beta =1-{\sin }^2\beta$

$\Rightarrow {\cos }^2\beta =1-\frac{144}{x^2}$

$\Rightarrow \cos \beta =\sqrt{1-\frac{144}{x^2}}$

$\Rightarrow \beta ={\cos }^{-1}\frac{\sqrt{x^2-144}}{x}$

Thus ${\sin }^{-1}\frac{5}{x}+{\cos }^{-1}\frac{\sqrt{x^2-144}}{x}=\frac{\pi }{2}$

$\Rightarrow {\sin }^{-1}\frac{5}{x}=\frac{\pi }{2}-{\cos }^{-1}\frac{\sqrt{x^2-144}}{x}$

$\Rightarrow \frac{5}{x}=\sin (\frac{\pi }{2}-{\cos }^{-1}\frac{\sqrt{x^2-144}}{x})$

We know that $\sin (\frac{\pi }{2}-\theta )=\cos \theta$

$\Rightarrow \frac{5}{x}=\cos \left({\cos }^{-1}\frac{\sqrt{x^2-144}}{x}\right)$

$\Rightarrow \frac{5}{x}=\frac{\sqrt{x^2-144}}{x}$

$\Rightarrow \sqrt{x^2-144}=5$

$\Rightarrow x^2-144=5^2$

$\Rightarrow x^2-144=25$

$\Rightarrow x^2=144+25$

$\Rightarrow x^2=169$

$\Rightarrow x=13$