Answer:
[tex]tan(\theta)=-\frac{5}{\sqrt{119}}[/tex]
Step-by-step explanation:
The correct question is
Suppose that ∅ Is an angle with csc(∅)=-12/5 and ∅ Is not in the third quadrant. Compute the exact value of Tan(∅).
∅ Is not in the third quadrant
If csc(∅) is negative the angle lie in the III Quadrant or in the IV Quadrant
∅ Is not in the third quadrant ----> given problem
so
That means ----> ∅ Is in the fourth quadrant
step 1
Find the value of [tex]sin(\theta)[/tex]
we have
[tex]csc(\theta)=-\frac{12}{5}[/tex]
we know that
[tex]csc(\theta)=\frac{1}{sin(\theta)}[/tex]
therefore
[tex]sin(\theta)=-\frac{5}{12}[/tex]
step 2
Find the value of [tex]cos(\theta)[/tex]
we know that
[tex]sin^2(\theta)+cos^2(\theta)=1[/tex]
we have
[tex]sin(\theta)=-\frac{5}{12}[/tex]
substitute
[tex](-\frac{5}{12})^2+cos^2(\theta)=1[/tex]
[tex]\frac{25}{144}+cos^2(\theta)=1[/tex]
[tex]cos^2(\theta)=1-\frac{25}{144}[/tex]
[tex]cos^2(\theta)=\frac{119}{144}[/tex]
[tex]cos(\theta)=\frac{\sqrt{119}}{12}[/tex] ---> is positive (IV Quadrant)
step 3
Find the value of [tex]tan(\theta)[/tex]
we know that
[tex]tan(\theta)=\frac{sin(\theta)}{cos(\theta)}[/tex]
substitute the values
[tex]tan(\theta)=-\frac{5}{12} : \frac{\sqrt{119}}{12}=-\frac{5}{\sqrt{119}}[/tex]
