Question

If ${\sin }^{-1}\left(\frac{3}{5}\right)+{\sin }^{-1}\left(\frac{8}{17}\right)={\cos }^{-1}\left(\frac{a}{b}\right)$.
Find the value of $a$ and $b$.

• A. $a=+35,b=85$
• B. $a=-36,b=85$
• C. $a=+77,b=85$
• D. $a=-77,b=85$

Answer 1

Answer: (A)

$a=+35,b=85$

Given, ${\sin }^{-1}\left(\frac{3}{5}\right)+{\sin }^{-1}\left(\frac{8}{17}\right)={\cos }^{-1}\left(\frac{a}{b}\right)$
$\Rightarrow {\sin }^{-1}(\frac{3}{5}\sqrt{1-{\left(\frac{8}{17}\right)}^2}+\frac{8}{17}\sqrt{1-{\left(\frac{3}{5}\right)}^2})={\cos }^{-1}\frac{a}{b}$
$\Rightarrow {\sin }^{-1}(\frac{3}{5}\times \frac{15}{17}+\frac{8}{17}\times \frac{4}{5})={\cos }^{-1}\frac{a}{b}$
$\Rightarrow {\sin }^{-1}\left(\frac{77}{85}\right)={\cos }^{-1}\left(\frac{a}{b}\right)$
$\Rightarrow {\cos }^{-1}\left(\sqrt{1-{\left(\frac{77}{85}\right)}^2}\right)={\cos }^{-1}\left(\frac{a}{b}\right)$
$\Rightarrow {\cos }^{-1}\left(\frac{35}{85}\right)={\cos }^{-1}\left(\frac{a}{b}\right)\Rightarrow a=35,b=85$