Final answer:
To find tan(A-B), we first determine cos(A) and cos(B) using the given values for sin(A) and tan(B), respectively, and their relevant trigonometric identities. Then we apply the formula for tan of difference of angles to obtain tan(A-B).
Step-by-step explanation:
Given that sin(A) = 5/13, with A in the second quadrant (π/2 < A < π), we know that cos(A) is negative in this quadrant. To find cos(A), we use the Pythagorean identity sin²(A) + cos²(A) = 1. Substituting sin(A) gives us (5/13)² + cos²(A) = 1, which then allows us to solve for cos(A). We get cos(A) = -√(1 - (5/13)²), which simplifies to cos(A) = -12/13.
For tan(B) = -√13, with B also in the second quadrant (π/2 < B < π), we know that both sine and cosine of B are negative. The identity to find cos(B) from tan(B) is tan(B) = sin(B)/cos(B), which gives us sin(B) = -√13 × cos(B). To find cos(B), we know that sin²(B) + cos²(B) = 1, and since tan²(B) + 1 = 1/cos²(B), by substituting tan(B) we find cos(B) = -1/√(tan²(B) + 1) = -1/√((-√13)² + 1), which simplifies to cos(B) = -1/2.
We can then use the formula for the tan of the difference of two angles: tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B)). Substituting the values for tan(A) and tan(B) we have found, we get tan(A - B) = ((5/13)/(-12/13) - (-√13)/(-1/2)) / (1 + (5/13)(-√13)). Simplifying this expression gives us the final value of tan(A - B).