To find the exact value of sin(a+B), we need to use the trigonometric identities and the given information.
First, we can use the information about angle a and tangent to find the values of sin(a) and cos(a). Since angle a lies in quadrant 2 and tangent is negative, we know that cosine is positive in that quadrant. Therefore, cos(a) = β(1 - sin^2(a)) = β(1 - ((12/5)^2)) = β(1 - (144/25)) = β(25/25 - 144/25) = β(-119/25).
Next, we can use the information about angle B and cosine to find the value of sin(B). Since angle B lies in quadrant 4 and cosine is positive, we know that sine is negative in that quadrant. Therefore, sin(B) = -β(1 - cos^2(B)) = -β(1 - ((3/5)^2)) = -β(1 - (9/25)) = -β(25/25 - 9/25) = -β(16/25) = -β(4/5).
Now, we can use the sum-of-angle identity for sine to find sin(a+B):
sin(a+B) = sin(a)cos(B) + cos(a)sin(B).
Plugging in the values we found:
sin(a+B) = (β(-119/25))((3/5)) + (β(4/5))(-β(4/5)).
Simplifying further:
sin(a+B) = (-β(357)/25) + (-4/5).
Combining like terms:
sin(a+B) = (-β(357) - 4)/5.
Therefore, the exact value of sin(a+B) is (-β(357) - 4)/5.