If ` log_("sin"(pi)/(4)) sin x gt 0 , x in [ 0. 4pi)`. Them find the region of x satisfying the inequality .

Explanation

To solve the inequality \( \log_{\sin(\frac{\pi}{4})}(\sin x) > 0 \) for \( x \in [0, 4\pi) \), we will follow these steps:

### Step 1: Understand the logarithmic inequality
The inequality \( \log_{\sin(\frac{\pi}{4})}(\sin x) > 0 \) implies that \( \sin x \) must be greater than \( \sin(\frac{\pi}{4}) \) since the base \( \sin(\frac{\pi}{4}) \) is positive and less than 1.

### Step 2: Calculate \( \sin(\frac{\pi}{4}) \)
We know that:
\[
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