You are given that sin(A)=−16/65, with A in Quadrant III, and cos(B)=77/85, with B in Quadrant I. Find sin(A+B). Give your answer as a fraction.

Answer 1

Since sin A = -16/65 and π < A < 3π/2 then cos A = -( 1 - [16/65]² )^1/2= - 63/65

Also since cos B = 77/85 and 0 < B < π/2 then sin B = ( 1 - ( 77/85 )²)^1/2 = 36/85

Then sin ( A +B ) = sin A cos B + cos A sin B

= ( - 16/65 ) ( 77/85 ) + ( -63/65) ( 36/85) =

= - 5444/5525

Answer 2

A = arcsin(-16/65) = -14.25 deg => 180 deg +14.25 deg == 194.25 deg (Added 180 deg to get to Quad III)

B = arccos(77/85) = 25.06 deg

sin(A+B) = sin(194.25+25.06) = sin(219.31) = -0.633516

Answer => -63/100