Final answer:
To determine cos(u + v) with given sin u and sin v, u in quadrant IV and v in quadrant II, and given that cos(u + v) = 0, the exact value is 0 by directly using the provided conditions without the need for additional calculations.
Step-by-step explanation:
To find the exact value of cos(u + v), given that sin u = cos (u + v) = 0, and knowing that sin v is given with a value in quadrant II, and u is in quadrant IV, we need to apply trigonometric identities and properties of the unit circle. First, since u is in quadrant IV, cos u is positive and sin u is negative. Since sin u = 0, u must be an angle where the sine function equals zero, which would be at 0 or π; however, since u is in the fourth quadrant, u must be 2π (360 degrees).
For v in the second quadrant, cos v would be negative since sine is positive in that quadrant. Given sin v = √13 (we assume the question meant √(13) because only the square root of a non-negative value is a real number), we can use the Pythagorean identity sin² v + cos² v = 1 to find cos v. However, as sin u = 0, the entire expression for cos(u + v) will be zero because cos(u + v) is given as 0. Therefore, the exact value of cos(u + v) is 0.