Given cotA= 4/3 , tan B= - 7/24; π <A<3π/2 and π/2<B< π find sin(A-B)

Answer 1

Sin(A-b) = sinAcosB - cosAsinB

A is in quadrant 3 where sine and cosine are negative

tan A = 3/4 ---> opposite=-3 and adjacent=-4 ---> hypotenuse = 5

so sin A = -3/5 and cos A =-4/5

B is in quadrant 2, where sine is positive and cosine is negative

tan B = -7/24 ---> opposite = 7 and adjacent=-24 ---> hypotenuse = 25

sin B = 7/25 and cos B = -24/25

sin(A-b) = (-3/5)(-24/25) - (-4/5)(7/25)

= 72/125 + 28/125

100/125

= 4/5