Let's find the exact value of [tex]\(\csc\left(\frac{2\pi}{3}\right)\)[/tex].
First, recall the definition of the cosecant function:
[tex]\[
\csc(\theta) = \frac{1}{\sin(\theta)}
\][/tex]
Next, we need to determine [tex]\(\sin\left(\frac{2\pi}{3}\right)\)[/tex].
The angle [tex]\(\frac{2\pi}{3}\)[/tex] radians is equivalent to 120 degrees. We can use the unit circle or trigonometric identities to find the sine of this angle:
[tex]\[
\sin\left(120^\circ\right) = \sin\left(180^\circ - 60^\circ\right) = \sin\left(60^\circ\right)
\][/tex]
From the unit circle, we know that:
[tex]\[
\sin(60^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
So:
[tex]\[
\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}
\][/tex]
Now, we can find [tex]\(\csc\left(\frac{2\pi}{3}\right)\)[/tex]:
[tex]\[
\csc\left(\frac{2\pi}{3}\right) = \frac{1}{\sin\left(\frac{2\pi}{3}\right)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}
\][/tex]
To rationalize the denominator, multiply the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
\][/tex]
Therefore, the exact value of [tex]\(\csc\left(\frac{2\pi}{3}\right)\)[/tex] is:
[tex]\[
\frac{2\sqrt{3}}{3}
\][/tex]
The correct answer is:
[tex]\[
\frac{2 \sqrt{3}}{3}
\][/tex]
