Final answer:
To find tan(A-B), first find the values of cos(A) and cos(B) using the given information. Then use the trigonometric identities to calculate tan(A-B) using the values of sin(A), cos(A), tan(B), and cos(B).
Step-by-step explanation:
We are given that sin(A) = 5/13 and tan(B) = -√13. We need to find the value of tan(A-B).
First, let's find the values of cos(A) and cos(B) using the given information. Since sin(A) = 5/13, we can use the Pythagorean identity sin^2(A) + cos^2(A) = 1 to solve for cos(A). It becomes cos^2(A) = 1 - sin^2(A) = 1 - (5/13)^2 = 1 - 25/169 = 144/169. Taking the square root of both sides, we get cos(A) = ±12/13. Since π/2 < A < π, A is in the second quadrant, so cos(A) = -12/13.
Similarly, since tan(B) = -√13, we can use the identity tan^2(B) + 1 = sec^2(B) to solve for cos(B). It becomes cos^2(B) = 1 - tan^2(B) = 1 - (-√13)^2 = 1 - 13 = -12. Since π/2 < B < π, B is in the second quadrant, so cos(B) is negative. Taking the square root of both sides of cos^2(B) = -12, we get cos(B) = ±√12i. But since cos(B) is negative, we have cos(B) = -√12.
Next, we can use the trigonometric identities to find the value of tan(A-B). Using the identity tan(A-B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B)), we can substitute the known values: tan(A) = sin(A)/cos(A) = (5/13)/(-12/13) = -5/12 and tan(B) = -√13. Plugging these values into the formula, we get:
tan(A-B) = (-5/12 - (-√13))/(1 + (-5/12)(-√13)) = (-5/12 + √13)/(1 + 5√13/12)