ANSWER
sin(u/2) = 0.9333
EXPLANATION
To answer this question, we have to use the formula for the sine of the half angle,
[tex]\sin\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1-\cos\theta}{2}}[/tex]
In this case, the cosine of the angle is given, cos u = -0.742,
[tex]\sin\left(\frac{u}{2}\right)=\pm\sqrt{\frac{1-(-0.742)}{2}}=\pm\sqrt{\frac{1+0.742}{2}}=\pm\sqrt{\frac{1.742}{2}}=\pm\sqrt{0.871}\approx\pm0.9333[/tex]
How do we decide which sign to use? This is why the quadrant information is given. The angle, u, is in quadrant III, so the half angle must be either in quadrant I or in quadrant I. Angles in these two quadrants have a positive sine. Therefore, we have to use the positive result.
Hence, the result is sin(u/2) = 0.9333.
