Answer:
Part 1) [tex]csc(\theta)=-\frac{\sqrt{85}}{7}[/tex]
Part 2) [tex]sin(\theta)=-\frac{7}{\sqrt{85}}[/tex] or [tex]sin(\theta)=-\frac{7\sqrt{85}}{85}[/tex]
Part 3) [tex]tan(\theta)=-\frac{7}{6}[/tex]
Part 4) [tex]cos(\theta)=\frac{6}{\sqrt{85}}[/tex] or [tex]cos(\theta)=\frac{6\sqrt{85}}{85}[/tex]
Part 5) [tex]sec(\theta)=\frac{\sqrt{85}}{6}[/tex]
Step-by-step explanation:
we know that
The angle theta lie on the IV Quadrant
so
sin(θ) is negative
cos(θ) is positive
tan(θ) is negative
sec(θ) is positive
csc(θ) is negative
step 1
Find the value of csc(θ)
we know that
[tex]1+cot^{2}(\theta)=csc^{2}(\theta)[/tex]
we have
[tex]cot(\theta)=-\frac{6}{7}[/tex]
substitute
[tex]1+(-\frac{6}{7})^{2}=csc^{2}(\theta)[/tex]
[tex]1+\frac{36}{49}=csc^{2}(\theta)[/tex]
[tex]\frac{85}{49}=csc^{2}(\theta)[/tex]rewrite
[tex]csc(\theta)=-\frac{\sqrt{85}}{7}[/tex] ----> remember that is negative
step 2
Find the value of sin(θ)
we know that
[tex]csc(\theta)=\frac{1}{sin(\theta)}[/tex]
we have
[tex]csc(\theta)=-\frac{\sqrt{85}}{7}[/tex]
therefore
[tex]sin(\theta)=-\frac{7}{\sqrt{85}}[/tex]
or
[tex]sin(\theta)=-\frac{7\sqrt{85}}{85}[/tex]
step 3
Find the value of tan(θ)
we know that
[tex]tan(\theta)=\frac{1}{cot(\theta)}[/tex]
we have
[tex]cot(\theta)=-\frac{6}{7}[/tex]
therefore
[tex]tan(\theta)=-\frac{7}{6}[/tex]
step 4
Find the value of cos(θ)
we know that
[tex]sin^{2}(\theta)+cos^{2}(\theta)=1[/tex]
we have
[tex]sin(\theta)=-\frac{7}{\sqrt{85}}[/tex]
substitute
[tex](-\frac{7}{\sqrt{85}})^{2}+cos^{2}(\theta)=1[/tex]
[tex]\frac{49}{85}+cos^{2}(\theta)=1[/tex]
[tex]cos^{2}(\theta)=1-\frac{49}{85}[/tex]
[tex]cos^{2}(\theta)=\frac{36}{85}[/tex]
[tex]cos(\theta)=\frac{6}{\sqrt{85}}[/tex] ------> the cosine is positive
or
[tex]cos(\theta)=\frac{6\sqrt{85}}{85}[/tex]
step 5
Find the value of sec(θ)
we know that
[tex]sec(\theta)=\frac{1}{cos(\theta)}[/tex]
we have
[tex]cos(\theta)=\frac{6}{\sqrt{85}}[/tex]
therefore
[tex]sec(\theta)=\frac{\sqrt{85}}{6}[/tex] ----> is positive
