Find sin(A−B) given that cosA=1/3, with A in quadrant I, and sinB=−1/2, with B in quadrant IV.

Answer 1

sin(A - B) = sinAcosB - cosAsinB

sinA = √1 - 1/9) = 2√2/3; cosB = √1 - 1/4 = √3/2;

sin(A - B) = 2√2/3·√3/2 - 1/3(- 1/2) = √6/3 + 1/6

Answer 2

sin(A-B) = sinAcosB - cosAsinB

cosA^2 + sinA^2 = 1

(1/3)^2 + sinA^2 = 1

solve for sinA, positive since in quadrant I.

cosB^2 + sinB^2 = 1

cosB^2 + (-1/2)^2 = 1

solve for cosB, positive since in quadrant IV.

plug given values for cosA and sinB and computed values for sinA and cosB into difference formula.