Let $\theta$ be an angle such that $\tan \theta =\frac{3}{2}$ and $\csc \theta <0$. Find the exact values of $\sin \theta$ and $\cos \theta$.

Step 1 ：Let $\theta$ be an angle such that $\tan \theta =\frac{3}{2}$ and $\csc \theta <0$. We are to find the exact values of $\sin \theta$ and $\cos \theta$.

Step 2 ：The tangent of an angle is positive in the first and third quadrants, while the cosecant of an angle is negative in the third and fourth quadrants. Therefore, the angle $\theta$ must be in the third quadrant. In the third quadrant, both sine and cosine are negative.

Step 3 ：We know that $\tan \theta =\frac{\sin \theta }{\cos \theta }=\frac{3}{2}$. We can use the Pythagorean identity ${\sin }^2\theta +{\cos }^2\theta =1$ to find the exact values of $\sin \theta$ and $\cos \theta$.

Step 4 ：By solving the Pythagorean identity, we find that the exact values of $\sin \theta$ and $\cos \theta$ are $\boxed{{\displaystyle -\frac{3}{\sqrt{13}}}}$ and $\boxed{{\displaystyle -\frac{2}{\sqrt{13}}}}$, respectively.