Question

The value of $\theta \in (0,2\pi )$ for which the equation $2si{n}^2\theta -5sin\theta +2>0$ is

• A. $(0,\frac{\pi }{6})\cup (\frac{5\pi }{6},2\pi )$
  
• B. $(\frac{\pi }{8},\frac{5\pi }{6})$
  
• C. $(0,\frac{\pi }{8})\cup (\frac{\pi }{6},\frac{5\pi }{6})$
  
• D. $(\frac{41\pi }{48},\pi )$
  

Answer 1

The correct option is A $(0,\frac{\pi }{6})\cup (\frac{5\pi }{6},2\pi )$

$2si{n}^2\theta -5sin\theta +2>0$
or $(sin\theta -2)(2sin\theta -1)>0$
or $sin\theta <\frac{1}{2}[\because -1\le sin\theta \le 1]$

From the graph $x\in (0,\frac{\pi }{6})\cup (\frac{5\pi }{6},2\pi )$