Question

The inequation is satisfied $2{\sin }^2(x-\frac{\pi }{3})-5\sin (x-\frac{\pi }{3})+2>0$

• A. is satisfied in $(-\frac{5\pi }{6},\frac{5\pi }{6})$
• B. is satisfied in $(-\frac{5\pi }{6},\frac{\pi }{2})$
• C. is satisfied in $(-\frac{(12n-5)\pi }{6},\frac{(4n+1)\pi }{2}),n\in I$
• D. is satisfied for all x for which $sin(x-\frac{\pi }{3})<\frac{1}{2}$

Answer 1

The correct option is D is satisfied for all x for which $sin(x-\frac{\pi }{3})<\frac{1}{2}$

${\sin }^2(x-\frac{\pi }{3})-5\sin (x-\frac{\pi }{3})+2>0$

$\Rightarrow [\sin (x-\frac{\pi }{3})-2][2\sin (x-\frac{\pi }{3})-1]>0$

Since, $\sin (x-\frac{\pi }{3})<1$

$\Rightarrow \sin (x-\frac{\pi }{3})-2<0$

Therefore, $\sin (x-\frac{\pi }{3})<\frac{1}{2}$