Question

Find the value of $x\text{if}{\sin }^{-1}\left(1/3\right)+{\sin }^{-1}\left(\frac{2}{3}\right)={\sin }^{-1}x$

Answer 1

Given ${\text{sin}}^{-1}(\frac{1}{3})+{\text{sin}}^{-1}(\frac{2}{3})={\text{sin}}^{-1}x$

$x=\sin ({\text{sin}}^{-1}\left(1/3\right)+{\text{sin}}^{-1}\left(2/3\right))$

$⟹x=\sin \left({\text{sin}}^{-1}\right(1/3\left)\right)\cos \left({\text{sin}}^{-1}\right(2/3\left)\right)+\cos \left({\text{sin}}^{-1}\right(1/3\left)\right)\sin \left({\text{sin}}^{-1}\right(2/3\left)\right)$

$⟹x=\frac{1}{3}\times \sqrt{1-\frac{4}{9}}+\frac{2}{3}\times \sqrt{1-\frac{1}{9}}=\frac{\sqrt{5}+4\sqrt{2}}{9}$