Question

Решить тригонометрическое уравнение:

Answer 1

$\begin{cases}2\sin x\cos x=\sin 2x\\ \sin x+\cos x=\sqrt{2}\cos\left ( x-\cfrac{\pi}{4} \right )\end{cases}\Rightarrow 5\sin 2x-12\sqrt{2}\cos \left ( x-\cfrac{\pi}{4} \right )+5=0\\t=x-\cfrac{\pi}{4}\Rightarrow 5\sin \left ( 2t+\cfrac{\pi}{2} \right )-12\sqrt{2}\cos t+5=0\Leftrightarrow 10\cos^2t-12\sqrt{2}\cos t=0\\\cos t\underbrace{\left ( 10\cos t-12\sqrt{2} \right )}_{t\in\mathbb{C}}=0\Rightarrow t=\cfrac{\pi}{2}+\pi k,k \in\mathbb{Z}\Rightarrow x=\cfrac{3\pi}{4}+\pi k,k \in\mathbb{Z}$