Question 174442
It is a little difficult to decipher what you are asking here.  I think you mean find the exact value of {{{sin(u-v)}}} given that {{{sin(u)=5/13}}} and {{{cos(v)=-3/5}}}


First of all, you need the difference identity:  {{{sin(alpha-beta)=sin(alpha)cos(beta)-cos(alpha)sin(beta)}}}.


Your problem is that you now need to determine {{{cos(u)}}} given that {{{sin(u)=5/13}}} and {{{sin(v)}}} given that {{{cos(v)=-3/5}}}.


Knowing that {{{sin(u)>0}}}, you are certain that {{{u}}} is an angle in the First or Second Quadrants.  Remembering that a 5-12-13 triangle is a right triangle and the fact that the sine function is defined as opposite/hypotenuse, we can see directly that {{{cos(u)=12/13}}} (if {{{u}}} is First Quadrant) or {{{cos(u) = -12/13}}} (if {{{u}}} is Second Quadrant).


Similarly, {{{v}}} must be Quadrant II or III because {{{cos(v)<0}}} and 3-4-5 is a right triangle, so {{{sin(v)=4/5}}} or {{{sin(v)=-4/5}}}


All you need to do is plug the values into {{{sin(alpha-beta)=sin(alpha)cos(beta)-cos(alpha)sin(beta)}}} or, in the case of this problem {{{sin(u-v)=sin(u)cos(v)-cos(u)sin(v)}}}, and do the arithmetic.  Because there are two possible values for each of the derived single angle functions, you will have 4 different calculations to perform, however you will end up with two pairs of identical results -- hence you will have two answers in the end.