Question

Solve for $x$ :
${\log }_2\left(sin\frac{x}{2}\right)<-1$

• A. $-\frac{\pi }{3}$
• B. $\frac{5\pi }{3}$
• C. $\frac{2\pi }{3}$
• D. $\frac{\pi }{3}$

Answer 1

The correct option is D $\frac{\pi }{3}$

${\log }_2\left(\sin \frac{x}{2}\right)<-1$

$\Rightarrow \sin \frac{x}{2}<2^{-1}$ using basic definition of logarithm.

$\Rightarrow \sin \frac{x}{2}<\frac{1}{2}$ since $a^{-m}=\frac{1}{a^m}$

$\sin \theta =\frac{1}{2}$ when $\theta =\frac{\pi }{6}$

$\therefore \frac{x}{2}=\frac{\pi }{6}$

Hence $x=\frac{2\pi }{6}=\frac{\pi }{3}$