Question

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

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

Answer 1

The correct option is C $a=+77,b=85$

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