## To find sin(a - B), we need to use the trigonometric identities and information given to break down the expression step-by-step. Let's go through the process:

Step 1: First, find sin(B) using the given information:

Since cos(B) = 15/17 and B lies in quadrant I, we know that sin(B) is positive. To find sin(B), we can use the Pythagorean identity: sin^2(B) + cos^2(B) = 1.

Plugging in the value of cos(B), we have:

sin^2(B) + (15/17)^2 = 1

sin^2(B) + 225/289 = 1

sin^2(B) = 1 - 225/289

sin^2(B) = (289 - 225)/289

sin^2(B) = 64/289

Taking the square root of both sides, we get:

sin(B) = Â±8/17

Since B is in quadrant I and sin(B) is positive, we have sin(B) = 8/17.

Step 2: Next, find cos(a) using the given information:

We know that sin(a) = 12/13, and a is in quadrant II. In quadrant II, cos(a) is negative. We can use the Pythagorean identity to find cos(a):

sin^2(a) + cos^2(a) = 1

(12/13)^2 + cos^2(a) = 1

144/169 + cos^2(a) = 1

cos^2(a) = 1 - 144/169

cos^2(a) = (169 - 144)/169

cos^2(a) = 25/169

Taking the square root of both sides and considering that cos(a) is negative in quadrant II, we have:

cos(a) = -5/13

Step 3: Finally, use the trigonometric addition formula to find sin(a - B):

The trigonometric addition formula for sin is: sin(a - B) = sin(a) * cos(B) - cos(a) * sin(B)

Plugging in the values we found earlier, we have:

sin(a - B) = (12/13) * (15/17) - (-5/13) * (8/17)

sin(a - B) = (180/221) + (40/221)

sin(a - B) = 220/221

Therefore, sin(a - B) = 220/221.