Giải phương trình: cos(2x+π/4)+cos(2x−π/4)+4sinx=2+√2(1−sinx)

Hướng dẫn giải:

(*) \( \Leftrightarrow 2\cos 2x.\cos \frac{\pi }{4}+4\sin x=2+\sqrt{2}(1-\sin x) \)

 \( \Leftrightarrow \sqrt{2}(1-2{{\sin }^{2}}x)+(4+\sqrt{2})\sin x-2-\sqrt{2}=0\Leftrightarrow 2\sqrt{2}{{\sin }^{2}}x-(4+\sqrt{2})\sin x+2=0 \)

 \( \Leftrightarrow 2{{\sin }^{2}}x-(2\sqrt{2}+1)\sin x+\sqrt{2}=0\Leftrightarrow \left[ \begin{align}  & \sin x=\sqrt{2}\text{ }(\ell ) \\  & \sin x=\frac{1}{2} \\ \end{align} \right. \)

 \( \Leftrightarrow x=\frac{\pi }{6}+k2\pi \vee x=\frac{5\pi }{6}+k2\pi ,\text{ }k\in \mathbb{Z} \).