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

To calculate $\tan\left(\frac{4\pi}{3}\right)$, we start by locating the angle $\frac{4\pi}{3}$ on the unit circle.

The angle $\frac{4\pi}{3}$ radians is equivalent to $240^\circ$.

This angle lies in the third quadrant where both sine and cosine are negative.

Using the unit circle, we find the coordinates of the point at $240^\circ$: $(\cos 240^\circ, \sin 240^\circ) = \left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right)$.

The tangent of an angle is given by the ratio of the sine to the cosine:

$\tan\left(\frac{4\pi}{3}\right) = \frac{\sin 240^\circ}{\cos 240^\circ} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}.$

To determine $ anleft(frac{4pi}{3}
ight)$, first find the reference angle.

The reference angle for $frac{4pi}{3}$ is $pi – frac{4pi}{3} = frac{pi}{3}$.

The coordinates for the reference angle $frac{pi}{3}$ are $left(frac{1}{2}, frac{sqrt{3}}{2}
ight)$.

In the third quadrant, the coordinates for $frac{4pi}{3}$ change signs: $left(-frac{1}{2}, -frac{sqrt{3}}{2}
ight)$.

The tangent is:

$ anleft(frac{4pi}{3}
ight) = frac{sinleft(frac{4pi}{3}
ight)}{cosleft(frac{4pi}{3}
ight)} = frac{-frac{sqrt{3}}{2}}{-frac{1}{2}} = sqrt{3}.$

Compute $ anleft(frac{4pi}{3}
ight)$:

$ anleft(frac{4pi}{3}
ight) = frac{sinleft(frac{4pi}{3}
ight)}{cosleft(frac{4pi}{3}
ight)}.$

With $sinleft(frac{4pi}{3}
ight) = -frac{sqrt{3}}{2}$ and $cosleft(frac{4pi}{3}
ight) = -frac{1}{2}$,

$ anleft(frac{4pi}{3}
ight) = frac{-frac{sqrt{3}}{2}}{-frac{1}{2}} = sqrt{3}.$

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