Question

3. Решить уравнение arcsin(x)+arcsin(x/2)=pi/4 

Answer 1

$arcsin(x)+arcsin( frac{x}{2})= frac{ pi}{4}$
$x=sin(a); x/2=sin(b);$
$left { {{a+b= frac{ pi}{4} } atop {sin(a)=2*sin(b)}} right.$
$sin(a)=2sin(frac{ pi }{4}-a)$
$sin(a)= sqrt{2}(cos(a)-sin(a)$)
$frac{1}{ sqrt{2} } =ctg(x)-1$
$ctg(a)= frac{sqrt{2}-2 }{2}$
$frac{ +-sqrt{1- x^{2}} }{x} = frac{sqrt{2}-2 }{2}$
$4-4 x^{2} =6 x^{2} -4 sqrt{2} x^{2}$
$x^{2} = frac{4}{10-4 sqrt{2} }$
$x = +- sqrt{frac{4}{10-4 sqrt{2} } }$