Answer:
Part 1) [tex]sin(\theta)=-\frac{\sqrt{23}}{5}[/tex]
Part 2) [tex]tan(\theta)=\frac{\sqrt{46}}{2}[/tex]
Step-by-step explanation:
step 1
Find the [tex]sin(\theta)[/tex]
we know that
[tex]sin^{2}(\theta) +cos^{2}(\theta)=1[/tex]
we have
[tex]cos(\theta)=-\frac{\sqrt{2}}{5}[/tex]
substitute
[tex]sin^{2}(\theta) +(-\frac{\sqrt{2}}{5})^{2}=1[/tex]
[tex]sin^{2}(\theta) +\frac{2}{25}=1[/tex]
[tex]sin^{2}(\theta)=1-\frac{2}{25}[/tex]
[tex]sin^{2}(\theta)=\frac{23}{25}[/tex]
square root both sides
[tex]sin(\theta)=\pm\frac{\sqrt{23}}{5}[/tex]
Remember that the angle θ terminates in Quadrant III
That means, that the value of sin(θ) is negative
so
[tex]sin(\theta)=-\frac{\sqrt{23}}{5}[/tex]
step 2
Find the [tex]tan(\theta)[/tex]
we know that
[tex]tan(\theta)=\frac{sin(\theta)}{cos(\theta)}[/tex]
we have
[tex]sin(\theta)=-\frac{\sqrt{23}}{5}[/tex]
[tex]cos(\theta)=-\frac{\sqrt{2}}{5}[/tex]
substitute
[tex]tan(\theta)=-\frac{\sqrt{23}}{5}:-\frac{\sqrt{2}}{5}=\frac{\sqrt{23}}{\sqrt{2}}[/tex]
simplify
[tex]tan(\theta)=\frac{\sqrt{46}}{2}[/tex]
