The value of tan(u/2) is -0.492.
We are given that sec(u) = -1.372 and u is in quadrant 3. We are asked to find tan(u/2).
We can use the following identity to find tan(u/2) from sec(u):
tan(u/2) = (sin(u))/(1 + cos(u))
First, we need to find sin(u) and cos(u). We know that sec(u) = 1/cos(u), so we can solve for cos(u) as follows:
cos(u) = 1/sec(u) = 1/(-1.372) = -0.729
We can use the Pythagorean identity to find sin(u) as follows:
sin^2(u) + cos^2(u) = 1
sin^2(u) = 1 - cos^2(u) = 1 - (-0.729)^2 = 0.682
sin(u) = sqrt(sin^2(u)) = sqrt(0.682) = 0.824
Now that we know sin(u) and cos(u), we can plug them into the identity to find tan(u/2):
tan(u/2) = (sin(u))/(1 + cos(u)) = (0.824)/(1 - 0.729) = -0.492
Therefore, tan(u/2) = -0.492.