You are given that sin(A)=−13/85, with A in Quadrant III, and cos(B)=56/65, with B in Quadrant I. Find sin(A+B). Give your answer as a fraction.

Question

give answer as a fraction

Answer 1

sin(A + B) = sinA*cosB + cosA*sinB

sin(A) = opp/hypotenus = -13/85

by Pythagorean Theorem adjacent side = sqrt(85^2 - 13^2) = 84

cos(A) = adjacent/hypotenuse = -84/85 (since in quadrant 3)

cos(B) = adjacent/hypotenuse = 56/65

by Pythagorean Theorem opposite side = sqrt(65^2 - 56^2) = 33

sin(B) = opposite/hypotenuse = 33/65

Plug values for sin and cos of A and B into equation to get final answer.

Answer 2

The person creating this problem had fun using Pythagorean triples many would not know -

33-56-65 and 13-84-85

In quadrant 3, sin and cos are negative. Thus, since sin(A) = -13/85, cos(A) = -84/85

In quadrant 1, sin and cos are positive. Since cos(B) = 56/65, sin(B) = 33/65

sin(A+B) = sin(A)cos(B) + cos(A)sin(B) = -13/85(56/65) -84/85(33/65) = -140/221