4. using the unit circle, calculate the value of csc(4π/3). -2√3/3 √3 -√3/2 √3/3

Explanation:

Step1: Recall the definition of cosecant

$\csc(x)=\frac{1}{\sin(x)}$. So we need to find $\sin(\frac{4\pi}{3})$.

Step2: Find the reference - angle

The angle $\theta=\frac{4\pi}{3}$ is in the third - quadrant. The reference angle $\theta_{r}=\frac{4\pi}{3}-\pi=\frac{\pi}{3}$.

Step3: Determine the sign of sine in the third - quadrant

In the third - quadrant, $\sin(x)<0$. And $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$, so $\sin(\frac{4\pi}{3})=-\frac{\sqrt{3}}{2}$.

Step4: Calculate the cosecant value

Since $\csc(\frac{4\pi}{3})=\frac{1}{\sin(\frac{4\pi}{3})}$, substituting $\sin(\frac{4\pi}{3}) = -\frac{\sqrt{3}}{2}$, we get $\csc(\frac{4\pi}{3})=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2\sqrt{3}}{3}$.

Answer:

$\frac{-2\sqrt{3}}{3}$ (corresponding to the first option)