see the picture attached to better understand the problem
we know that
in the right triangle ABC
tan A=opposite side angle A/adjacent side angle A
tan A=4/3
opposite side angle A=BC
adjacent side angle A=AB
tan A=BC/AB
so
4/3=BC/AB----> 4*AB=3*BC-----> equation 1
sin C=3/5
sin C=opposite side angle C/hypotenuse
opposite side angle C=AB
hypotenuse=AC
so
sin C=AB/AC
3/5=AB/AC-----> 3*AC=5*AB
AC=15
so
3*15=5*AB--------> AB=9
substitute the value of AB in equation 1
4*AB=3*BC-----> 4*9=3*BC-----> BC=12
the answer is
BC=12
