Find the exact value of the trigonometric function given that sin u=3/5 and cos v=-8/17. Both u and v are in quadrant II. Tan (u+v)

To find the value of tan (u+v), we'll need to use the sum formula for tangent:

tan (u+v) = (tan u + tan v) / (1 - tan u * tan v)

First, let's find the values of tan u and tan v.

Given that sin u = 3/5, we can use the fact that sine is the opposite over the hypotenuse in a right triangle to find the value of the adjacent side.
Using the Pythagorean theorem, we can calculate the length of the adjacent side:
a^2 + (3/5)^2 = 1^2
a^2 + 9/25 = 1
a^2 = 1 - 9/25
a^2 = 16/25
Taking the square root on both sides, we can find the value of a:
a = √(16/25)
a = 4/5

Since u is in the second quadrant, the adjacent side is negative. Thus, tan u = sin u / cos u is negative:
tan u = -3/4

Given cos v = -8/17, we can again use the Pythagorean theorem to find the value of the adjacent side:
(-8/17)^2 + b^2 = 1^2
64/289 + b^2 = 1
b^2 = 289/289 - 64/289
b^2 = 225/289
Taking the square root on both sides, we can find the value of b:
b = √(225/289)
b = 15/17

Since v is in the second quadrant, the adjacent side is positive. Thus, tan v = sin v / cos v is negative:
tan v = -15/8

Now we can substitute the values of tan u and tan v into the sum formula for tangent:

tan (u+v) = (tan u + tan v) / (1 - tan u * tan v)
tan (u+v) = (-3/4 + (-15/8)) / (1 - (-3/4)(-15/8))
tan (u+v) = (-3/4 - 15/8) / (1 - 45/32)
tan (u+v) = (-6/8 - 15/8) / (32/32 - 45/32)
tan (u+v) = (-21/8) / (-13/32)
tan (u+v) = (-21/8) * (-32/13)
tan(u+v) = 672/104
tan(u+v) = 168/26
tan(u+v) = 84/13

Therefore, the exact value of tan(u+v) is 84/13.

use Pythagoras (a^2 + b^2 = c^2) to find the angle functions

then use the trig identity to find the tangent