[tex]\displaystyle\\\text{If }~~180^o<\theta<270^o~~\text{then }~~\theta\in~\text{quadrant 3}\\\\\text{In the 3rd cotangent dial is positive.}\\\\\text{We use the formula: } ~~~\boxed{1+\cot^2\theta=\csc^2\theta}[/tex]
[tex]\displaystyle\\1+\cot^2\theta=\csc^2\theta\\\\\cot^2\theta=\csc^2\theta-1\\\\\cot^2\theta=\left(-\frac{3}{2}\right)^2-1\\\\\\\cot^2\theta=\left(\frac{3}{2}\right)^2-1\\\\\\\cot^2\theta=\frac{9}{4}-\frac{4}{4}\\\\\\\cot^2\theta=\frac{5}{4}\\\\\\\cot\theta=\pm\sqrt{\frac{5}{4}}\\\\\\\text{We will eliminate the negative solution.}\\\\\\\cot\theta=+\sqrt{\frac{5}{4}}\\\\\\\boxed{\bf\cot\theta=\frac{\sqrt{5}}{2}}}[/tex]
