Giải phương trình: sin2x+cos2x+7sinx-cosx-4=0

* Hướng dẫn giải

Lời giải:

$$\begin{gathered} \begin{array}{l}pt\iff \sin 2x+1-2{\sin }^{2}x+\text{7sinx}-\cos x-4=0\\ \iff \sin 2x-\cos x-2{\sin }^{2}x+\text{7sinx}-3=0\\ \iff 2\sin x\cos x-\cos x-(2\sin x-1)(\text{sinx+2)}=0\\ \iff \cos x(2\sin x-1)-(2\sin x-1)(\text{sinx+2)}=0\\ \iff (2\sin x-1)(\cos x-\text{sinx}-\text{2)}=0\end{array} \\ \iff \left[\begin{array}{l}2\sin x-1=0\\ \cos x-\text{sinx}-\text{2=0}\end{array}\right.\iff \left[\begin{array}{l}\sin x=\frac{1}{2}\\ \text{sin}\left(\text{x-}\frac{\pi }{4}\right)-\sqrt{2}\text{=0}\end{array}\right. \\ \iff \left[\begin{array}{l}\sin x=\frac{1}{2}\\ \text{sin}\left(\text{x-}\frac{\pi }{4}\right)=\sqrt{2}\left(vn\right)\end{array}\right.\iff \sin x=\sin \frac{\pi }{6}\iff \left[\begin{array}{l}x=\frac{\pi }{6}+k2\pi \\ x=\frac{5\pi }{6}+k2\pi \end{array}\right. \end{gathered}$$