Calculate the value of $ anleft(frac{4pi}{3}
ight)$ on the unit circle

First, let’s understand the position of $\frac{4\pi}{3}$ on the unit circle. The angle $\frac{4\pi}{3}$ radians is in the third quadrant.

In the third quadrant, the reference angle is $\frac{\pi}{3}$. The tangent is positive in the third quadrant.

We know that $\tan\left(\frac{\pi}{3}\right) = \sqrt{3}$. Therefore:

$ \tan\left(\frac{4\pi}{3}\right) = \tan\left(\pi + \frac{\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} $

The angle $frac{4pi}{3}$ is in the third quadrant of the unit circle. The reference angle for $frac{4pi}{3}$ is $frac{pi}{3}$.

In the third quadrant, the tangent function is positive. Therefore:

$ anleft(frac{4pi}{3}
ight) = anleft(pi + frac{pi}{3}
ight) = anleft(frac{pi}{3}
ight) $

We know that $ anleft(frac{pi}{3}
ight) = sqrt{3}$. Hence:

$ anleft(frac{4pi}{3}
ight) = sqrt{3} $

The angle $frac{4pi}{3}$ lies in the third quadrant where the tangent function is positive.

The reference angle is $frac{pi}{3}$.

Therefore:

$ anleft(frac{4pi}{3}
ight) = anleft(frac{pi}{3}
ight) = sqrt{3} $

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