Final answer:
To find cos(u+v), determine cos u and cos v using the trigonometric identities and given sine values, and then apply the cosine addition formula cos(u+v) = cos u * cos v - sin u * sin v.
Step-by-step explanation:
The question involves finding cos(u+v) given that sin u = -3/5 with u in quadrant III, and sin v = 12/13 with v in quadrant II. In these quadrants, cosine is negative for both angles as well. Using the trigonometric identity for cosine of a sum, cos(u+v) = cos u * cos v - sin u * sin v, we need to find cos u and cos v first.
Since sin u = -3/5, we can use the Pythagorean identity, 1 = sin² u + cos² u, to find cos u. Similarly, cos v can be found using 1 = sin² v + cos² v. After finding the values of cos u and cos v, we can substitute into cos(u+v) to find the desired result.