Final answer:
Using the cosine difference identity and the provided sine and cosine values, along with the Pythagorean identity, the exact value of cos(u − v) is found to be − 84/85.
Step-by-step explanation:
To find the exact value of the trigonometric expression cos(u − v), we can use the cosine difference identity along with the given information that sin(u) = − 3/5 and cos(v) = 15/17.
The cosine difference identity is cos(u − v) = cos(u)cos(v) + sin(u)sin(v). Since we know sin(u) and cos(v), we need to find cos(u) and sin(v) in order to apply the identity.
Given that sin(u) = − 3/5 and 3π/2 < u < 2π, we are in the fourth quadrant where cosine is positive. So we can find cos(u) using the Pythagorean identity cos^2(u) + sin^2(u) = 1:
- cos^2(u) = 1 − sin^2(u) = 1 − (− 3/5)^2 = 1 − 9/25 = 16/25
- cos(u) = − √(16/25) = − 4/5 (negative because u is in the fourth quadrant)
Similarly, since cos(v) = 15/17 and 0 < v < π/2, we are in the first quadrant where sine is positive. Using the Pythagorean identity again:
- sin^2(v) = 1 − cos^2(v) = 1 − (15/17)^2 = 1 − 225/289 = 64/289
- sin(v) = √(64/289) = 8/17
Now, using the identity and substituting the values, we get the exact value of cos(u − v):
- cos(u − v) = cos(u)cos(v) + sin(u)sin(v)
- cos(u − v) = (− 4/5)(15/17) + (− 3/5)(8/17)
- cos(u − v) = − 60/85 − 24/85
- cos(u − v) = − 84/85
Therefore, the exact value of cos(u − v) is − 84/85.