Question

Find the value of $x$ for which
$\sin h^{-1}\frac{3}{4}+\sin h^{-1}x=\sin h^{-1}\frac{4}{3}$

Answer 1

from the formula,

$\sin h^{-1}x=\ln (x+\sqrt{x^2+1})$

substitute the above formula in given equation

$\ln \left(3/4\sqrt{(3/4{)}^2+1}\right)+\ln (x+\sqrt{x^2+1})=\ln (4/3+\sqrt{(4/3{)}^2+1})$

$\Rightarrow \ln \left(2\right)+\ln \ast x+\sqrt{x^2+1})=\ln (3)$

$\Rightarrow \ln (x+\sqrt{x^2+1})=\ln \left(3\right)-\ln \left(2\right)(\ln A-\ln B=\ln (A/B)$

$\Rightarrow x+\sqrt{x^2+1}=3/2$

$\Rightarrow x^2+1=(3/2-x{)}^2$

$\Rightarrow x=5/12$