To determine the exact value of [tex]\(\tan \left(\frac{2\pi}{3}\right)\)[/tex], let's follow these steps carefully:
1. We recognize that [tex]\(\frac{2\pi}{3}\)[/tex] is an angle in radians. To better understand where this angle lies, we can convert it to degrees or visualize it on the unit circle.
[tex]\[
\frac{2\pi}{3} \text{ radians} = 120^\circ
\][/tex]
2. The angle [tex]\(120^\circ\)[/tex] is in the second quadrant. Trigonometric functions have specific signs in each quadrant:
- In the second quadrant, sine is positive and cosine is negative.
- Tangent, being the ratio of sine to cosine, is negative in the second quadrant.
3. We use the reference angle to determine [tex]\(\tan \left(\frac{2\pi}{3}\right)\)[/tex]. The reference angle for [tex]\(120^\circ\)[/tex] is [tex]\(180^\circ - 120^\circ = 60^\circ\)[/tex] or [tex]\(\pi - \frac{2\pi}{3} = \frac{\pi}{3}\)[/tex].
4. We know the tangent of the reference angle:
[tex]\[
\tan \left(\frac{\pi}{3}\right) = \sqrt{3}
\][/tex]
5. Given that tangent is negative in the second quadrant:
[tex]\[
\tan \left( \frac{2\pi}{3} \right) = -\tan \left( \frac{\pi}{3} \right) = -\sqrt{3}
\][/tex]
Therefore, the exact value of [tex]\(\tan \left( \frac{2\pi}{3} \right) \)[/tex] is:
[tex]\[
-\sqrt{3}
\][/tex]
So, the correct answer is:
[tex]\[
-\sqrt{3}
\][/tex]
