sin(A-B) = 33/65
cos(2A) = -41/145
Finding sin(A-B):
To find sin(A-B), we can use the formula:
sin(A-B) = sin A * cos B - cos A * sin B
Given that sin A = -20/29 and cos B = 12/13, we need to determine cos A and sin B. Since A is in quadrant III, cos A is negative. Using the Pythagorean identity sin^2 A + cos^2 A = 1, we find cos A = -21/29. For B in quadrant I, sin B is positive. Using the same identity, sin B = 5/13.
Now substitute these values into the formula:
sin(A-B) = (-20/29) * (12/13) - (-21/29) * (5/13)
After simplifying, we get sin(A-B) = 33/65.
Finding cos(2A):
The formula for cos(2A) is:
cos(2A) = cos^2 A - sin^2 A
We already know sin A and cos A. Substitute these values into the formula:
cos(2A) = (-21/29)^2 - (-20/29)^2
After simplifying, we get cos(2A) = -41/145.