Giải phương trình: sin^2 x - 3sinxcosx + 1 = 0

Đề bài: Giải phương trình: ${\sin }^{2}x-3\sin x.\cos x+1=0$

Trả lời

Hướng dẫn giải:

$\begin{array}{l}{\sin }^{2}x-3\sin x.\cos x+1=0\\ \iff {\sin }^{2}x-3\sin x.\cos x+{\sin }^{2}x+{\cos }^{2}x=0\\ \iff 2{\sin }^{2}x-2\sin x.\cos x+co{s}^{2}x-\sin x.\cos x=0\\ \iff 2\sin x\left(\mathrm{s}\text{inx}-\cos x\right)+\cos x\left(\cos x-\sin x\right)=0\\ \iff \left(\sin x-\cos x\right)\left(2\sin x-\cos x\right)=0\\ \iff \left[\begin{array}{l}\sin x-\cos x=0\\ 2\sin x-\cos x=0\end{array}\right.\iff \left[\begin{array}{l}\sqrt{2}.cos\left(x+\frac{\pi }{4}\right)=0\\ \frac{2}{\sqrt{5}}\sin x-\frac{1}{\sqrt{5}}\cos x=0\end{array}\right.\end{array}$

$\begin{array}{l}\iff \left[\begin{array}{l}\cos \left(x+\frac{\pi }{4}\right)=0\\ sin\left(x+\alpha \right)=0\left(\alpha =\arcsin \frac{1}{\sqrt{5}}\right)\end{array}\right.\\ \iff \left[\begin{array}{l}x=\frac{\pi }{4}+k\pi \\ x=\alpha +k\pi \end{array}\right.\left(k\in \mathbb{Z}\right)\end{array}$