Because the angle omega is in QIII, the adjacent side lies to the left of the origin and thus is represented by a negative value. Since sin omega = -3/5, the opposite side is also negative. Use the Pythagorean Theorem to obtain the length of the adjacent side: 5^2 = (-3)^2 + x^2, or 25 = 9 + x^2, or x^2 = 16.
Solving for x: x = plus or minus 4. Since we are in QIII, let x = -4.
Then cos omega is opp / hyp, or -4/5.