Remember that
[tex]\tan\theta=\dfrac{\sin\theta}{\cos\theta}[/tex]
So if [tex]\sin\theta>0[/tex] and [tex]\tan\theta<0[/tex], then we know [tex]\cos\theta<0[/tex].
Then use the Pythagorean identity to solve for cosine:
[tex]\cos^2\theta+\sin^2\theta=1\implies\cos\theta=-\sqrt{1-\sin^2\theta}=-\dfrac{2\sqrt{10}}7[/tex]
