Phương trình -2*cos^2(x) - 5*sinx +4=0 có bao nhiêu nghiệm thuộc đoạn [0;9*pi/2]

Phương trình $-2{\cos }^{2}x-5\sin x+4=0\iff -2\left(1-{\sin }^{2}x\right)-5\sin x+4=0$

$\iff 2{\sin }^{2}x-5\sin x+2=0\iff \left[\begin{array}{l}\sin x=2\\ \sin x=\frac{1}{2}\end{array}\right.\Rightarrow \sin x=\frac{1}{2}=\sin \frac{\pi }{6}$

$\iff \left[\begin{array}{l}x=\frac{\pi }{6}+k2\pi \\ x=\frac{5\pi }{6}+l2\pi \end{array}\right.\left(k,l\in \mathbb{Z}\right)$.

Vì $x\in \left[0;\frac{9\pi }{2}\right]$ nên $\left[\begin{array}{l}0\le \frac{\pi }{6}+k2\pi \le \frac{9\pi }{2}\\ 0\le \frac{5\pi }{6}+l2\pi \le \frac{9\pi }{2}\end{array}\right.\iff \left[\begin{array}{l}-\frac{1}{6}\le 2k\le \frac{9}{2}-\frac{1}{6}\\ -\frac{5}{6}\le 2l\le \frac{9}{2}-\frac{5}{6}\end{array}\right.\iff \left[\begin{array}{l}-\frac{1}{12}\le k\le \frac{13}{6}\\ -\frac{5}{12}\le l\le \frac{11}{6}\end{array}\right.$

$\Rightarrow \left[\begin{array}{l}k\in \left\{0;1;2\right\}\\ l\in \left\{0;1\right\}\end{array}\right.$. Vậy phương trình có 5 nghiệm.