First of all, sinx = 7/5 is not possible because the opposite side (7) should not be longer than the hypotenuse (5), unless this should have been sinx = 5/7.
So, instead if we just use sinx = 5/7 in quadrant II:
x = arcsin(5/7) = 45.58 degrees (this is the reference angle), so
x = 180 - 45.58 = 134.42 degrees
Then cos y = 12/13 in quadrant I:
y = arccos(12/13) = 22.62 degrees
Therefore,
sin (x + y) = sinx cosy + cosx siny
= sin 134.42 cos 22.62 + cos 134.42 sin 22.62
= (0.7142)(0.9231) + (-0.6999)(0.3846)
= 0.6593 - 0.2692
= 0.3901