If $\cos (x)=-\frac{14}{23}$ (in Quadrant-111), find Give exact answers. \[ \sin \left(\frac{x}{2}\right)= \]

Step 1 ：Given that \(\cos (x)=-\frac{14}{23}\) (in Quadrant-111), we are asked to find the exact value of \(\sin \left(\frac{x}{2}\right)\).

Step 2 ：We know that the cosine function is negative in the third quadrant, so the sine function will also be negative in this quadrant.

Step 3 ：We can use the half-angle formula for sine, which is \(\sin \left(\frac{x}{2}\right)= \pm \sqrt{\frac{1-\cos(x)}{2}}\). Since we are in the third quadrant, we will use the negative root.

Step 4 ：Substituting the given value of \(\cos(x)\) into the formula, we get \(\sin \left(\frac{x}{2}\right)= -\sqrt{\frac{1-(-\frac{14}{23})}{2}}\).

Step 5 ：Solving this, we get \(\sin \left(\frac{x}{2}\right)\) to be approximately -0.8968544062928813.

Step 6 ：So, the exact value of \(\sin \left(\frac{x}{2}\right)\) is \(\boxed{-0.8968544062928813}\).