We know that:
[tex]\sin (\theta_1)=-\frac{12}{13}[/tex]
There is also an interesting property that relates the sine and the cosine of an angle:
[tex]\sin ^2(\theta_1)+\cos ^2(\theta_1)=1[/tex]
We can find the cosine of theta using this equation:
[tex]\begin{gathered} \cos ^2(\theta_1)=1-\sin ^2(\theta_1) \\ \cos (\theta_1)=\sqrt{1-\sin^2(\theta_1)} \\ \cos (\theta_1)=\sqrt[]{1-(-\frac{12}{13})^2} \\ \lvert\cos (\theta_1)\rvert=\sqrt[]{1-\frac{144}{169}}=\sqrt[]{\frac{25}{169}} \\ \lvert\cos (\theta_1)\rvert=\frac{5}{13} \end{gathered}[/tex]
Since theta is in the third quadrant then its cosine must be a negative number so:
[tex]\cos (\theta_1)=-\frac{5}{13}[/tex]
