Answer:
[tex]\sin\theta=-\dfrac{\sqrt{65}}{9}.[/tex]
[tex]\tan\theta=-\dfrac{\sqrt{65}}{4}.[/tex]
Step-by-step explanation:
1. Use the man trigonometric equality
[tex]\cos^2\theta+\sin^2\theta=1.[/tex]
From this equality
[tex]\sin^2\theta=1-\cos^2\theta,\\ \\\sin^2\theta=1-\left(\dfrac{4}{9}\right)^2,\\ \\\sin^2\theta=1-\dfrac{16}{81},\\ \\\sin^2\theta=\dfrac{65}{81}.[/tex]
2. Since [tex]\csc\theta=\dfrac{1}{\sin\theta}<0,[/tex] you can state that [tex]\sin\theta<0[/tex] and
[tex]\sin\theta=-\sqrt{\dfrac{65}{81}}=-\dfrac{\sqrt{65}}{9}.[/tex]
3. Use the definition:
[tex]\tan\theta=\dfrac{\sin\theta}{\cos\theta}=-\dfrac{\frac{\sqrt{65}}{9}}{\frac{4}{9}}=-\dfrac{\sqrt{65}}{4}.[/tex]
