Answer:
Part 1) [tex]sin(\theta)=\frac{2}{3}[/tex]
Par 2) [tex]cos(\theta)=-\frac{\sqrt{5}}{3}[/tex]
Part 3) [tex]tan(\theta)=-\frac{2\sqrt{5}}{5}[/tex]
Step-by-step explanation:
step 1
Find the [tex]sin(\theta)[/tex]
we have
[tex]csc(\theta)=\frac{3}{2}[/tex]
Remember that
[tex]csc(\theta)=\frac{1}{sin(\theta)}[/tex]
therefore
[tex]sin(\theta)=\frac{2}{3}[/tex]
step 2
Find the [tex]cos(\theta)[/tex]
we know that
[tex]sin^{2}(\theta) +cos^{2}(\theta)=1[/tex]
we have
[tex]sin(\theta)=\frac{2}{3}[/tex]
substitute
[tex](\frac{2}{3})^{2} +cos^{2}(\theta)=1[/tex]
[tex]\frac{4}{9} +cos^{2}(\theta)=1[/tex]
[tex]cos^{2}(\theta)=1-\frac{4}{9}[/tex]
[tex]cos^{2}(\theta)=\frac{5}{9}[/tex]
square root both sides
[tex]cos(\theta)=\pm\frac{\sqrt{5}}{3}[/tex]
we have that
[tex]cos(\theta) < 0[/tex] ---> given problem
so
[tex]cos(\theta)=-\frac{\sqrt{5}}{3}[/tex]
step 3
Find the [tex]tan(\theta)[/tex]
we know that
[tex]tan(\theta)=\frac{sin(\theta)}{cos(\theta)}[/tex]
we have
[tex]sin(\theta)=\frac{2}{3}[/tex]
[tex]cos(\theta)=-\frac{\sqrt{5}}{3}[/tex]
substitute
[tex]tan(\theta)=\frac{2}{3}:-\frac{\sqrt{5}}{3}=-\frac{2}{\sqrt{5}}[/tex]
Simplify
[tex]tan(\theta)=-\frac{2\sqrt{5}}{5}[/tex]
