Use the given information to evaluate sin (a-b). Exact answers. No decimals.

Sin a = -7/25, cot b = 8/15; neither a nor b are in quadrant III.

since no QIII angles, a is in QIV and b is in QI

sina = -7/25
cosa = 24/25

sinb = -15/17
cosb = -8/17

now just evaluate

sin(a-b) = sina cosb - cosa sinb

To evaluate sin (a-b), we can make use of the trigonometric identities. One of the identities that will be useful for this problem is the sine of the difference of two angles identity:

sin (a-b) = sin a * cos b - cos a * sin b

Given that sin a = -7/25 and cot b = 8/15, we will need to find the values of cos a and sin b to substitute into the formula.

To find cos a, we can use the Pythagorean identity:

sin^2 a + cos^2 a = 1

Substituting sin a = -7/25:

(-7/25)^2 + cos^2 a = 1
49/625 + cos^2 a = 1
cos^2 a = 1 - 49/625
cos^2 a = 576/625
cos a = ± √(576/625)
cos a = ± 24/25

Since neither a nor b are in quadrant III, both sine and cosine will have the same sign. Therefore, cos a = 24/25.

To find sin b, we can use the fact that cot b is the reciprocal of tan b:

cot b = 8/15
tan b = 15/8

Using the Pythagorean identity:

tan^2 b + 1 = sec^2 b

Substituting tan b = 15/8:

(15/8)^2 + 1 = sec^2 b
225/64 + 1 = sec^2 b
sec^2 b = (225 + 64)/64
sec^2 b = 289/64
sec b = ± √(289/64)
sec b = ± 17/8

Since cot b is positive, sec b must also be positive. Therefore, sec b = 17/8.

We can now substitute the values into the sine of the difference of two angles formula:

sin (a-b) = sin a * cos b - cos a * sin b
sin (a-b) = (-7/25) * (17/8) - (24/25) * (8/15)
sin (a-b) = (-7/25 * 17/8) - (24/25 * 8/15)
sin (a-b) = (-119/200) - (192/375)
sin (a-b) = (-119 * 375 - 192 * 200)/(200 * 375)
sin (a-b) = (-44625 - 38400)/(75000)
sin (a-b) = -83025/75000

Therefore, sin (a-b) = -83025/75000.