Answer::
![\begin{gathered} \sin (\theta)=\frac{\sqrt[]{5}}{3},\textcolor{red}{tan(\theta)=-\frac{\sqrt[]{5}}{2},}\cot (\theta)=-\frac{2\sqrt[]{5}}{5} \\ \sec (\theta)=-(3)/(2),\textcolor{red}{\csc (\theta)=\frac{3\sqrt[]{5}}{5}} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/wiq0lrgxk5899iponpmegn7zokoqb3322t.png)
Explanation:
Given:

If the cosine and tangent of an angle are both negative, then the angle is in Quadrant II.
First, determine the length of the opposite side using the Pythagorean Theorem.

The length of the opposite side is √5.
Therefore:
![\begin{gathered} \sin (\theta)=\frac{\text{Opposite}}{\text{Hypotenuse}}=\frac{\sqrt[]{5}}{3} \\ \tan (\theta)=\frac{\text{Opposite}}{\text{Adjacent}}=-\frac{\sqrt[]{5}}{2} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/m5zmej4fkgpzlkjm03hwkda8zixl9w0e8t.png)
Cotangent is the inverse of tangent, therefore:
![\begin{gathered} \cot (\theta)=(1)/(\tan(\theta))=-\frac{2}{\sqrt[]{5}} \\ \text{Rationalise the denominator} \\ =-\frac{2}{\sqrt[]{5}}*\frac{\sqrt[]{5}}{\sqrt[]{5}} \\ \implies\cot (\theta)=-\frac{2\sqrt[]{5}}{5} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/klaengszrhgl3z6idgywzy4yi8exdcoioo.png)
Secant is the inverse of Cosine, therefore:

Cosecant is the inverse of Sine, therefore:
![\begin{gathered} \csc (\theta)=(1)/(\sin(\theta))=\frac{3}{\sqrt[]{5}} \\ \text{Rationalise the denominator} \\ =\frac{3}{\sqrt[]{5}}*\frac{\sqrt[]{5}}{\sqrt[]{5}} \\ \implies\csc (\theta)=\frac{3\sqrt[]{5}}{5} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/mnmknrc1epy5zfpn8ab6bzpb8to3qs85ju.png)