Question

The number of distinct solutions of the equation, $lo{g}_{\frac{1}{2}}\left|sinx\right|=2-lo{g}_{\frac{1}{2}}\left|cosx\right|$ in the interval $\left[0,2\pi \right]$, is__________.

Answer 1

Answer: 8.00

$lo{g}_{1/2}\left|sinx\right|=2-lo{g}_{1/2}\left|cosx\right|;x\in \left[0,2\pi \right]$
$\Rightarrow lo{g}_{1/2}\left|sinx\right|+lo{g}_{1/2}\left|cosx\right|=2$
$\Rightarrow lo{g}_{1/2}\left|sinxcosx\right|=2$
$\left|sinxcosx\right|=\frac{1}{4}\Rightarrow \left|sin2x\right|=\frac{1}{2}$
$\Rightarrow 8$ solutions