If log _{1/sqrt{2}} sin x 0 for x in [0, 4p | vedclass

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(D) Given the inequality $\log _{1/\sqrt{2}} \sin x > 0$. Since the base $b = \frac{1}{\sqrt{2}}$ satisfies $0 < b < 1$,the inequality reverses when w

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(D) Given the inequality $\log _{1/\sqrt{2}} \sin x > 0$. Since the base $b = \frac{1}{\sqrt{2}}$ satisfies $0 $\sin x $\sin x Also,for the logarithm to be defined,we must have $\sin x > 0$. Thus,we need $0 This condition is satisfied for all $x \in [0, 4\pi]$ except where $\sin x = 0$ or $\sin x = 1$. In the interval $[0, 4\pi]$,$\sin x = 0$ at $x = 0, \pi, 2\pi, 3\pi, 4\pi$. In the interval $[0, 4\pi]$,$\sin x = 1$ at $x = \frac{\pi}{2}, \frac{5\pi}{2}$. We are looking for values of $x$ that are integer multiples of $\frac{\pi}{4}$,i.e.,$x = k \cdot \frac{\pi}{4}$ for $k \in \{0, 1, 2, ..., 16\}$. The possible values are $x \in \{0, \frac{\pi}{4}, \frac{2\pi}{4}, \frac{3\pi}{4}, \frac{4\pi}{4}, \frac{5\pi}{4}, \frac{6\pi}{4}, \frac{7\pi}{4}, \frac{8\pi}{4}, \frac{9\pi}{4}, \frac{10\pi}{4}, \frac{11\pi}{4}, \frac{12\pi}{4}, \frac{13\pi}{4}, \frac{14\pi}{4}, \frac{15\pi}{4}, \frac{16\pi}{4}\}$. Excluding values where $\sin x \le 0$ or $\sin x = 1$: $x=0, \pi, 2\pi, 3\pi, 4\pi$ (where $\sin x = 0$) are excluded. $x=\frac{\pi}{2}, \frac{5\pi}{2}$ (where $\sin x = 1$) are excluded. The remaining values are $\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}, \frac{9\pi}{4}, \frac{11\pi}{4}, \frac{13\pi}{4}, \frac{15\pi}{4}$. There are $8$ such values.

If $\log _{1/\sqrt{2}} \sin x > 0$ for $x \in [0, 4\pi]$,find the number of values of $x$ that are integer multiples of $\frac{\pi}{4}$.

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