Question

Find the exact value of tan0 if csc0=-4/3and and the terminal side of 0 lies in Quadrant III.

Answer 1

D. tan(θ) = 3√7/7

Step-by-step explanation:

csc(θ) is equal to the hypotenuse over the opposite side, so if csc(θ) = -4/3, we can represent the angle with the following triangle

So, we can calculate the missing side using the Pythagorean theorem as follows

`\begin{gathered} x=\sqrt[]{4^2-3^2} \\ x=\sqrt[]{16-9} \\ x=\sqrt[]{7} \end{gathered}`

Now, the tangent of the angle is calculated as:

`\begin{gathered} \tan (\theta)=\frac{Opposite\text{ side}}{Adjacent\text{ side}} \\ \tan (\theta)=\frac{3}{\sqrt[]{7}} \end{gathered}`

Then, the tangent is equal to:

`\tan (\theta)=\frac{3}{\sqrt[]{7}}\cdot\frac{\sqrt[]{7}}{\sqrt[]{7}}=\frac{3\sqrt[]{7}}{7}`

Therefore, the answer is

D. tan(θ) = 3√7/7