Question

The number of distinct solutions of the equation ${\log }_{\frac{1}{2}}\left|\sin x\right|=2-{\log }_{\frac{1}{2}}\left|\cos x\right|$ in the interval $[0,2\mathrm{\pi }]$ is

Answer 1

Find the distinct solution of the given equation in the given range.

An equation ${\log }_{\frac{1}{2}}\left|\sin x\right|=2-{\log }_{\frac{1}{2}}\left|\cos x\right|$ is given and $x\in [0,2\mathrm{\pi }]$.

Simplify the given equation as follows:

$$\begin{gathered} {\log }_{\frac{1}{2}}\left|\sin x\right|+{\log }_{\frac{1}{2}}\left|\cos x\right|=2 \\ \Rightarrow {\log }_{\frac{1}{2}}\left|\sin x·\cos x\right|=2 \end{gathered}$$

We know that, $a^{{\log }_a\left(b\right)}=b$

Raise both sides to the exponent of $\frac{1}{2}$.

$$\begin{gathered} \left|\sin x·\cos x\right|={\left(\frac{1}{2}\right)}^2 \\ \Rightarrow \left|\sin x·\cos x\right|=\frac{1}{4} \\ \Rightarrow \left|2\sin x·\cos x\right|=\frac{1}{2} \\ \Rightarrow \left|\sin 2x\right|=\frac{1}{2} \\ \Rightarrow \sin 2x=\pm \frac{1}{2} \end{gathered}$$

For $\sin 2x=\pm \frac{1}{2}$..

$$\begin{gathered} 2x=\frac{\mathrm{\pi }}{3},\frac{5\mathrm{\pi }}{3},\frac{7\mathrm{\pi }}{6},\frac{11\mathrm{\pi }}{6},\frac{13\mathrm{\pi }}{6},\frac{17\mathrm{\pi }}{6},\frac{19\mathrm{\pi }}{6},\frac{23\mathrm{\pi }}{6},..... \\ \Rightarrow x=\frac{\mathrm{\pi }}{6},\frac{5\mathrm{\pi }}{6},\frac{7\mathrm{\pi }}{12},\frac{11\mathrm{\pi }}{12},\frac{13\mathrm{\pi }}{12},\frac{17\mathrm{\pi }}{12},\frac{19\mathrm{\pi }}{12},\frac{23\mathrm{\pi }}{12},..... \end{gathered}$$

All the other angles will be more than $2\mathrm{\pi }$.

Therefore, the total number of distinct solutions for the given equation in the interval $[0,2\mathrm{\pi }]$ is $8$.