Find the exact value of the trigonometric function given that

sin u = − 8/17 and cos v = − 3/5.
(Both u and v are in Quadrant III.)

cot(v − u)

cot(v-u) = (cotv cotu + 1)/(cotu - cotv)

= [(3/4)(15/8) + 1] / [(15/8) - (3/4)]
= (77/32) / (3/8)
= 77/12

I answered this question for a test and I got 40/43 as the cot since i did tan=(-4/5+15/8) and solved to get 43/40 and since cot is the reciprocal of tan I just flipped it over to get 40/43

To find the value of cot(v - u), we first need to find the values of cot(u) and cot(v).

Since we are given sin(u) and cos(v), we can find the values of sin(v) and cos(u) using the identities:
sin(v) = √(1 - cos^2(v))
cos(u) = √(1 - sin^2(u))

In this case, sin(u) = -8/17, so we can find cos(u) as follows:
cos(u) = √(1 - sin^2(u))
= √(1 - (-8/17)^2)
= √(1 - 64/289)
= √(289/289 - 64/289)
= √(225/289)
= 15/17

And cos(v) = -3/5, so we can find sin(v) as follows:
sin(v) = √(1 - cos^2(v))
= √(1 - (-3/5)^2)
= √(1 - 9/25)
= √(25/25 - 9/25)
= √(16/25)
= 4/5

Now we can find cot(u) and cot(v) using the reciprocals of sin and cos:
cot(u) = cos(u) / sin(u)
= (15/17) / (-8/17)
= -15/8

cot(v) = cos(v) / sin(v)
= (-3/5) / (4/5)
= -3/4

Finally, we can find cot(v - u) by subtracting cot(u) from cot(v):
cot(v - u) = cot(v) - cot(u)
= -3/4 - (-15/8)
= -3/4 + 15/8
= (-3*2 + 15)/8
= 9/8

Therefore, the exact value of cot(v - u) is 9/8.

To find the exact value of cot(v - u), we need to determine the values of cot(v) and cot(u), and then use the subtraction formula for cotangent.

Given that cos(v) = -3/5, we can determine cos(u) using the identity cos^2(u) = 1 - sin^2(u). Plugging in the value of sin(u) = -8/17, we get:

cos^2(u) = 1 - (-8/17)^2
cos^2(u) = 1 - 64/289
cos^2(u) = (289 - 64)/289
cos^2(u) = 225/289
cos(u) = ±√(225/289)
Since u is in Quadrant III, where cosine is negative, we take the negative square root:
cos(u) = -15/17

Using the definition of cotangent, cot(x) = cos(x) / sin(x), we can find cot(v) and cot(u):

cot(v) = cos(v) / sin(v)
cot(v) = (-3/5) / (-8/17)
cot(v) = (3/5) * (17/8)
cot(v) = 51/40

cot(u) = cos(u) / sin(u)
cot(u) = (-15/17) / (-8/17)
cot(u) = (15/17) * (17/8)
cot(u) = 15/8

Now, using the subtraction formula for cotangent, cot(v - u) = cot(v) * cot(u):

cot(v - u) = (51/40) * (15/8)
cot(v - u) = 765/320
cot(v - u) = 153/64

Therefore, the exact value of cot(v - u) is 153/64.