By definition, sin(phi) is an ordinate (Y-coordinate) of a unit vector positioned at angle angle phi counterclockwise from the X-axis, while cos(phi) is its abscissa (X-coordinate).
Obviously, sin^2(phi)+cos^2(phi)=1.
Since sin(alpha)=12/13, cos^2(alpha)=1-(12/13)^2=25/169 and, therefore, cos(alpha)=+-5/13 (correspondingly to an angle in the 1st quadrant and in the second one).
Since cos(beta)=-4/5, sin^2(beta)=1-(4/5)^2=9/25 and, therefore, sin(beta)=+-3/5 (correspondingly to an angle in the second quadrant and in the third one).
So, we have the following four combinations of values for sin and cos of angles alpha and beta:
(a) sin(alpha)=12/13, cos(alpha)=5/13, sin(beta)=3/5, cos(beta)=-4/5
(b) sin(alpha)=12/13, cos(alpha)=-5/13, sin(beta)=3/5, cos(beta)=-4/5
(c) sin(alpha)=12/13, cos(alpha)=5/13, sin(beta)=-3/5, cos(beta)=-4/5
(d) sin(alpha)=12/13, cos(alpha)=-5/13, sin(beta)=-3/5, cos(beta)=-4/5
Now we can use the formula
sin(alpha+beta)=sin(alpha)*cos(beta)+cos(alpha)*sin(beta)
(a) sin(alpha+beta)=12/13 * (-4/5) + 5/13 * 3/5=-33/65
(b) sin(alpha+beta)=12/13 * (-4/5) + (-5/13) * 3/5=-63/65
(c) sin(alpha+beta)=12/13 * (-4/5) + 5/13 * (-3/5)=-63/65
(d) sin(alpha+beta)=12/13 * (-4/5) + (-5/13) * (-3/5)=-33/65
So, we have two solutions:
sin(alpha+beta) = -33/65 and sin(alpha+beta) = -63/65.
