Question

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

Answer 1

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