Question

Simplify: ${\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{8}{17}$

Answer 1

${\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{8}{17}$

Using,

$si{n}^{-1}x+si{n}^{-1}y=sin-1(x\sqrt{1-y^2}+y\sqrt{1-x^2})$

$={\sin }^{-1}\{\frac{3}{5}\sqrt{1-{\left(\frac{8}{17}\right)}^2}+\frac{8}{17}\sqrt{1-{\left(\frac{3}{5}\right)}^2}\}$

$={\sin }^{-1}\{\frac{3}{5}\times \frac{15}{17}+\frac{8}{17}\times \frac{4}{5}\}$

$={\sin }^{-1}\left\{\frac{45+32}{85}\right\}={\sin }^{-1}\frac{77}{85}$