[tex]\sin \theta = \dfrac 2 3[/tex]
[tex]\tan \theta < 0[/tex]
Positive sine, negative tangent, means we have a negative cosine. We're talking about the second quadrant.
[tex]\cos^2 \theta + \sin ^2 \theta = 1[/tex]
[tex]\cos^2 \theta = 1 - \sin ^2\theta[/tex]
[tex]\cos \theta = \pm \sqrt{1 - \sin ^2\theta}[/tex]
We know it's negative,
[tex]\cos \theta = - \sqrt{1 - (2/3)^2} =-\sqrt{5/9} = - \dfrac 1 3 \sqrt{5}[/tex]
Answer: -(1/3)√5
