The identity for #sin ( alpha + beta )# is:
#sin ( alpha + beta )= sin(alpha)cos(beta)+cos(alpha)sin(beta)#
We are given the values #sin(alpha) = 3/5# and #cos(beta) = - 40/41#:
#sin ( alpha + beta )= (3/5)(-40/41)+cos(alpha)sin(beta)" [1]"#
We need to compute #cos(alpha) and sin(beta)#; for the former, use the identity:
#cos(alpha) = +-sqrt(1-sin^2(alpha))#
Substitute #sin^2(alpha) = (3/5)^2#
#cos(alpha) = +-sqrt(1-(3/5)^2)#
#cos(alpha) = +-sqrt(25/25-9/25)#
#cos(alpha) = +-sqrt(16/25)#
#cos(alpha) = +-4/5#
Because we are told that #alpha# is in the second quadrant, we choose the negative value:
#cos(alpha) = -4/5#
Substitute this into equation [1]:
#sin ( alpha + beta )= (3/5)(-40/41)+(-4/5)sin(beta)" [1.1]"#
Use the identity #sin(beta) = +-sqrt(1-cos^2(beta))#
Substitute #cos^2(beta)= (-40/41)^2#
#sin(beta) = +-sqrt(1-(-40/41)^2)#
#sin(beta) = +-sqrt(1-(-40/41)^2)#
#sin(beta) = +-sqrt(1681/1681-1600/1681)#
#sin(beta) = +-sqrt(81/1681)#
#sin(beta) = +-9/41#
Because we are told that #beta# is in the second quadrant, we choose the positive value:
#sin(beta) = 9/41#
Substitute into equation [1.1]
#sin ( alpha + beta )= (3/5)(-40/41)+(-4/5)(9/41)#
Perform the multiplication:
#sin ( alpha + beta )= -120/205-36/205#
#sin ( alpha + beta )= -156/205#
