ANSWER:
[tex]cot\: \theta=\: \frac{\sqrt[]{21}}{2}[/tex]
STEP-BY-STEP EXPLANATION:
We have the following information.
[tex]\begin{gathered} \sin \theta=-\frac{2}{5}=\frac{\text{ opposite}}{\text{ hypotenuse}} \\ \text{opposite = 2} \\ \text{ hypotenuse = 5} \end{gathered}[/tex]
We can calculate the adjacent leg by means of Pythagoras' theorem, just like this:
[tex]\begin{gathered} h^2=a^2+b^2 \\ 5^2=2^2+b^2 \\ b^2=25-4 \\ b=\sqrt[]{21} \\ \text{therefore} \\ \cos \theta=\frac{\text{adjacent}}{\text{hypotenuse}}=-\frac{\sqrt[]{21}}{5} \\ \text{replacing} \\ \cot \theta=\frac{\cos\theta}{\sin\theta}=\frac{-\frac{\sqrt[]{21}}{5}}{-\frac{2}{5}}=\frac{\sqrt[]{21}}{2} \\ \cot \theta=\frac{\sqrt[]{21}}{2} \\ \end{gathered}[/tex]
We determine that the angle is in the third quadrant
[tex]\begin{gathered} cot\: \theta=\: \frac{\sqrt{21}}{2} \\ \theta=\cot ^{-1}(\frac{\sqrt[]{21}}{2}) \\ \theta=203.58\text{\degree} \\ \text{ we check:} \\ \sin 203.58=-0.40002\cong-\frac{2}{5} \\ \cos 203.58=-0.91650\cong-\frac{\sqrt[]{21}}{5} \end{gathered}[/tex]
