The number of distinct solutions of the equation _1/2| x| = 2-_1/2| x| in the interval [0,…

Question

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

Answer: 8

Accepted Answer

$$\log_{1/2}|\sin x\cos x| = 2 \implies |\sin x\cos x| = \frac14 \implies |\sin2x| = \frac12$$

Over $[0,2\pi]$, $\sin2x=\pm\tfrac12$ has 8 solutions.

Solutions · 1