To determine the exact value of \(\csc \frac{2\pi}{3}\), we first recall that cosecant is the reciprocal of the sine function. Therefore, \(\csc x = \frac{1}{\sin x}\).
Let's find \(\sin \frac{2\pi}{3}\):
1. \(\frac{2\pi}{3}\) lies in the second quadrant. The reference angle for \(\frac{2\pi}{3}\) is \(\pi - \frac{2\pi}{3} = \frac{\pi}{3}\).
2. In the second quadrant, sine is positive. Recall that \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\).
3. Therefore, \(\sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}\).
Now, taking the reciprocal to find \(\csc \frac{2\pi}{3}\):
[tex]\[
\csc \frac{2\pi}{3} = \frac{1}{\sin \frac{2\pi}{3}} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}
\][/tex]
We can rationalize the denominator:
[tex]\[
\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
\][/tex]
So, the exact value of \(\csc \frac{2\pi}{3}\) is \(\frac{2\sqrt{3}}{3}\).
The correct answer is [tex]\(\frac{2 \sqrt{3}}{3}\)[/tex].
