Answer:
To calculate sin(A-B), use the sine difference identity with the given values for sinA and cosB, considering the correct signs for each value based on the quadrants where angles A and B are located.
Explanation:
To find sin(A-B), we can use the formula sin(A-B) = sinA cosB - cosA sinB. We already have cosA = 39/89 and sinB = -63/65. To find sinA and cosB, we can use the Pythagorean identity sin²A + cos²A = 1 and sin²B + cos²B = 1 since sin and cos are given.
First, we calculate sinA: sinA = √(1 - cos²A) = √(1 - (39/89)²) Since A is in Quadrant IV, sinA is negative, so sinA = -√((1 - (39/89)²).
Next, we calculate cosB: cosB = √(1 - sin²B) = √(1 - (-63/65)²) Since B is in Quadrant III (as inferred from the negative sine), cosB is also negative, so cosB = -√((1 - (-63/65)²).
Finally, substituting these values into the sin(A-B) formula:
sin(A-B) = (-√(1 - (39/89)²))(cosB) - (39/89)(-63/65) You can now calculate the numerical value using a calculator.