You are given that cos(A)=−725, with A in Quadrant III, and cos(B)=−1213, with B in Quadrant II. Find sin(A−B). Give your answer as a fraction.

Answer 1

With the information given, you need to first find the other two trig ratios that are required for the sine difference identity, which is: sin ( A - B ) = sin A cos B - cos A sin B. Setting up a diagram to represent both triangles would be helpful, and by Pythagorean Theorem, we see that the missing side in the triangle containing angle A is 24 and the missing side in the triangle containing angle B is 5. Setting up the appropriate trig ratios, keeping in mind the quadrants in which the trig ratios are positive and negative, we have sin A = -24/25 and sin B = 5/13. Now we insert all four trig ratio values into the sine difference identity:

sin ( A - B ) = ( -24/25)( -12/13) - ( -7/25)( 5/13 ). Now we simplify the expression and, since the denominators are already the same, we combine the numerators to end up with 323/325.