Question

решить уравнение arcsin 2x+arcsin x=п/3

Answer 1

$O.D.3.: left { {{-1 leq x leq 1} atop {-1 leq 2x leq 1}} right. => x in [-frac{1}{2};frac{1}{2}]\ arcsin2x=frac{pi}{3}-arcsin x\ sin(arcsin2x)=sin(frac{pi}{3}-arcsin x)$
$2x=sinfrac{pi}{3}cos(arcsin x)-cosfrac{pi}{3}sin(arcsin x)\ 2x=frac{sqrt3}{2}cos(arcsin x)-frac{1}{2}x\ 4x=sqrt3*sqrt{1-x^2}-x\ 3x=sqrt3*sqrt{1-x^2}$
$begin{cases} -frac{1}{2} leq x leq frac{1}{2} \ x geq 0 \ 1-x^2 geq 0 \ 9x^2=3(1-x^2) end{cases} => begin{cases} 0 leq x leq frac{1}{2} \ 4x^2=1 end{cases} => x= frac{1}{2}=0,5.$
Ответ: 0,5.