ANSWER:
[tex]-\frac{\sqrt[]{33}}{7}[/tex]STEP-BY-STEP EXPLANATION:
We have that being in the third quadrant, the angle is between 180 ° and 270 °.
We calculate the value of theta as follows
[tex]\begin{gathered} \sin \theta=-\frac{4}{7} \\ \theta=\sin ^{-1}(-\frac{4}{7})=-34.85 \\ \text{now, between 180\degree{}and 270\degree} \\ 180+34.85=214.85 \end{gathered}[/tex]Now knowing the angle, we can know the cosine of that angle
[tex]\cos 214.85=-0.82\cong-\frac{\sqrt[]{33}}{7}[/tex]