Something seems wrong to me. If \(\displaystyle \pi\) < x < \(\displaystyle \frac{3}{2}\pi\), then \(\displaystyle 2\pi\) < 2x < \(\displaystyle 3\pi\) and the sine is positive in that region.
So how could sin(2x) = -\(\displaystyle \frac{1}{3}\) AND \(\displaystyle \pi\) < x < \(\displaystyle \frac{3}{2}\pi\)?
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