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

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(A) Given the inequality $\log _{1/\sqrt{2}} \sin x > 0$.Since the base $1/\sqrt{2}$ is between $0$ and $1$,the inequality reverses when we remove the

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$4$

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(A) Given the inequality $\log _{1/\sqrt{2}} \sin x > 0$.Since the base $1/\sqrt{2}$ is between $0$ and $1$,the inequality reverses when we remove the logarithm:$0 $0 We need to find the values of $x = k \cdot \frac{\pi}{4}$ for $k \in \mathbb{Z}$ in the interval $[0, 4\pi]$ such that $0 The multiples of $\frac{\pi}{4}$ in $[0, 4\pi]$ are $0, \frac{\pi}{4}, \frac{2\pi}{4}, \frac{3\pi}{4}, \dots, \frac{16\pi}{4}$.Values where $\sin x = 0$ or $\sin x = 1$ must be excluded.For $x \in [0, 4\pi]$,$\sin x = 0$ at $x = 0, \pi, 2\pi, 3\pi, 4\pi$.$\sin x = 1$ at $x = \frac{\pi}{2}, \frac{5\pi}{2}$.Checking multiples of $\frac{\pi}{4}$:$x = \frac{\pi}{4} (\sin > 0), \frac{2\pi}{4} (\sin = 1, 	ext{exclude}), \frac{3\pi}{4} (\sin > 0), \frac{4\pi}{4} (\sin = 0, 	ext{exclude}), \frac{5\pi}{4} (\sin  0), \frac{10\pi}{4} (\sin = 1, 	ext{exclude}), \frac{11\pi}{4} (\sin > 0), \frac{12\pi}{4} (\sin = 0, 	ext{exclude}), \frac{13\pi}{4} (\sin The valid values are $\frac{\pi}{4}, \frac{3\pi}{4}, \frac{9\pi}{4}, \frac{11\pi}{4}$.There are $4$ such values.

If $\log _{1/\sqrt{2}} \sin x > 0$ for $x \in [0, 4\pi]$,then the number of values of $x$ which are integral multiples of $\frac{\pi}{4}$ is:

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