`sin(u + v)` Find the exact value of the trigonometric expression given that `sin(u) = 5/13` and `cos(v) = -3/5` (both `u` and `v` are in quadrant...

We know that `sin(u+v)=sin(u)cos(v)+cos(u)sin(v).`

`sin(u)` and `cos(v)` are given, let's find `cos(u)` and `sin(v).`

`cos(u)=+-sqrt(1-sin^2(u))=+-sqrt(1-25/169)=+-sqrt(144/169)=+-12/13.`

Select "-" because `u` is in quadrant II.

And `sin(v)=+-sqrt(1-cos^2(v))=+-sqrt(1-9/25)=+-4/5.`

Select "+" because `v` is in quadrant II.

Finally `sin(u+v)=(5/13)*(-3/5)+(-12/13)*(4/5)=-1/65*(15+48)=-63/65.`