Question

if cos(x)=(4/9) and tan(x)<0 then what is sin(x)?
Is it sin(x)=-(sqrt65)/9

Answer 1

Yes.

Answer 2

first you have a restrictment that tan(x) must be less than 0 meaning any negative number

Answer 3

it's negative in the second and 4th however cos is only positive in the fourth quadrant which makes it easy to figure the rest out