Question

Suppose that ? Is an angle with csc(?)=-12/5 and ? Is not in the third quadrant. Compute the exact value of Tan(?). You don’t have to rationalize the denominator.
I think the answer is -5/rad119, but I’m not sure

Answer 1

Answer:

$tan(\theta)=-\frac{5}{\sqrt{119}}$

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 $sin(\theta)$

we have

$csc(\theta)=-\frac{12}{5}$

we know that

$csc(\theta)=\frac{1}{sin(\theta)}$

therefore

$sin(\theta)=-\frac{5}{12}$

step 2

Find the value of $cos(\theta)$

we know that

$sin^2(\theta)+cos^2(\theta)=1$

we have

$sin(\theta)=-\frac{5}{12}$

substitute

$(-\frac{5}{12})^2+cos^2(\theta)=1$

$\frac{25}{144}+cos^2(\theta)=1$

$cos^2(\theta)=1-\frac{25}{144}$

$cos^2(\theta)=\frac{119}{144}$

$cos(\theta)=\frac{\sqrt{119}}{12}$ ---> is positive (IV Quadrant)

step 3

Find the value of $tan(\theta)$

we know that

$tan(\theta)=\frac{sin(\theta)}{cos(\theta)}$

substitute the values

$tan(\theta)=-\frac{5}{12} : \frac{\sqrt{119}}{12}=-\frac{5}{\sqrt{119}}$