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.
To evaluate sin (a - b), we can use trigonometric identities and the given information.
First, let's find the values of sin a and cos a using the given information. Since sin a = -7/25 and neither a nor b are in quadrant III, we know that sin a is negative. We can use the Pythagorean identity to find cos a.
Using the Pythagorean identity: sin^2(a) + cos^2(a) = 1
(-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
Next, let's find the value of cot b. We are given that cot b = 8/15. Since cot b is positive, we know that both cos b and sin b must also be positive.
Using the identity: cot^2(b) = 1/tan^2(b)
(8/15)^2 = 1/tan^2(b)
64/225 = 1/tan^2(b)
tan^2(b) = 225/64
tan b = √(225/64)
tan b = 15/8
Now, we have the values of sin a, cos a, and tan b. We can use the following identity to find sin (a - b):
sin (a - b) = sin a * cos b - cos a * sin b
Plugging in the values:
sin (a - b) = (-7/25)(√(1 - (15/8)^2)) - (24/25)(15/8)
Simplifying the expression:
sin (a - b) = (-7/25)(√(1 - 225/64)) - (24/25)(15/8)
sin (a - b) = (-7/25)(√(64/64 - 225/64)) - (24/25)(15/8)
sin (a - b) = (-7/25)(√(64 - 225)/64) - (24/25)(15/8)
sin (a - b) = (-7/25)(√(-161)/64) - (24/25)(15/8)
Since we are asked for an exact answer, we should leave the expression in this form as the final answer.