Final answer:
To find the exact value of sin(A + B), we can use the trigonometric identities for the sum of angles. Given that sin(A) = 4/5 and tan(B) = -5/12, we can calculate sin(B) using the Pythagorean identity. Finally, using the sum of angles identity, we can determine the exact value of sin(A + B) as -33/65.
Step-by-step explanation:
To find the exact value of sin(A + B), we can use the trigonometric identities for the sum of angles. Firstly, we need to find the values of sin(A) and sin(B). Given that sin(A) = 4/5 and tan(B) = -5/12, we can use the Pythagorean identity to find the value of sin(B). Since tan(B) = sin(B)/cos(B), we know that cos(B) = -12/13 (opposite sign since B is in Quadrant 2), and by using the Pythagorean identity, sin(B) = 5/13.
Now that we have sin(A) and sin(B), we can use the sum of angles identity, which states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Substituting the values we found: sin(A + B) = (4/5)(-12/13) + (3/5)(5/13) = -48/65 + 15/65 = -33/65.
Therefore, the exact value of sin(A + B) is -33/65.