Lets use # to represent tetha
B to represent beta
Given:
cos # = - root2/3
tan B = 4/3
we are to find sin (# + B)
cos # = - root2/3, which shows that # is in the second or third quadrant.
Adj = root 2, hyp = 3
Opp^2 = 3^2 - root 2^2
Opp^2 = 9 - 2
Opp = root 7
Therefore, sin # = root 7/3
Tan B = 4/3
Opp = 4, Adj = 3
Hyp^2 = 4^2 + 3^2
Hyp^2 = 25
Hyp = 5
Sin B = 4/5
Cos B = 3/5
Sin(# + B) = sin#cosB + sinBcos#
Sin(# + B) = root7/3 x 3/5 + 4/5 x -root 2/3
Sin(# + B) = root 7/5 - 4root 2/15