Answer:
[tex]\frac{5}{\sqrt{21}}[/tex].
Step-by-step explanation:
We are given [tex]\cos(\theta)=\frac{-2}{5}[/tex] and that [tex]\theta[/tex]'s angle terminates in quadrant 2 which means [tex](\cos(\theta)=\text{negative},\sin(\theta)=\text{positive})[/tex].
We will be using Pythagorean Identity: [tex]\sin^2(\theta)+\cos^2(\theta)=1[/tex].
[tex]\sin^2(\theta)+\cos^2(\theta)=1[/tex]
[tex]\sin^2(\theta)+(\frac{-2}{5})^2=1[/tex]
[tex]\sin^2(\theta)+\frac{4}{25}=1[/tex]
[tex]\sin^2(\theta)=1-\frac{4}{25}[/tex]
[tex]\sin^2(\theta)=\frac{25-4}{25}[/tex]
[tex]\sin^2(\theta)=\frac{21}{25}[/tex]
[tex]\sin(\theta)=\pm \sqrt{\frac{21}{25}}[/tex]
[tex]\sin(\theta)=\pm \frac{\sqrt{21}}{5}[/tex]
Since again [tex]\theta[/tex]'s angle terimates in quadrant 2, we will choose that [tex]\sin(\theta)=\frac{\sqrt{21}}{5}[/tex].
[tex]\csc(\theta)=\frac{1}{\sin(\theta)}=\frac{5}{\sqrt{21}}[/tex].
