QPaperGen

\(\textbf{Step 1: Determine quadrant of \(\frac{x}{2}\)}\)

Since \(x\) lies in the third quadrant, i.e., \(\pi < x < \frac{3\pi}{2}\), then:
\[
\frac{\pi}{2} < \frac{x}{2} < \frac{3\pi}{4}
\]
So, \(\frac{x}{2}\) lies in the second quadrant. Therefore:
\[
\sin\left(\frac{x}{2}\right) > 0,\quad \cos\left(\frac{x}{2}\right) < 0,\quad \tan\left(\frac{x}{2}\right) < 0
\]

\(\textbf{Step 2: Use half-angle identity for cosine}\)
\[
2\cos^2\left(\frac{x}{2}\right) = 1 + \cos x = 1 - \frac{1}{3} = \frac{2}{3}
\Rightarrow \cos^2\left(\frac{x}{2}\right) = \frac{1}{3}
\Rightarrow \cos\left(\frac{x}{2}\right) = -\frac{1}{\sqrt{3}}
\]

\(\textbf{Step 3: Use Pythagorean identity}\)
\[
\sin^2\left(\frac{x}{2}\right) = 1 - \cos^2\left(\frac{x}{2}\right) = 1 - \frac{1}{3} = \frac{2}{3}
\Rightarrow \sin\left(\frac{x}{2}\right) = \sqrt{\frac{2}{3}}
\]

\(\textbf{Step 4: Compute tangent}\)
\[
\tan\left(\frac{x}{2}\right) = \frac{\sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right)} = \frac{\sqrt{\frac{2}{3}}}{-\frac{1}{\sqrt{3}}} = -\sqrt{2}
\]

\(\textbf{Final Answer:}\)
\[
\sin\left(\frac{x}{2}\right) = \sqrt{\frac{2}{3}}, \quad
\cos\left(\frac{x}{2}\right) = -\frac{1}{\sqrt{3}}, \quad
\tan\left(\frac{x}{2}\right) = -\sqrt{2}
\]