Calculate $ an(frac{4pi}{3})$ using the Unit Circle

First, we need to find the reference angle for $\frac{4\pi}{3}$. The angle $\frac{4\pi}{3}$ is in the third quadrant.

The reference angle is $\pi – (\frac{4\pi}{3} – \pi) = \frac{\pi}{3}$.

In the third quadrant, the tangent function is positive, so we have:

$\tan(\frac{4\pi}{3}) = \tan(\frac{\pi}{3})$

$\tan(\frac{\pi}{3}) = \sqrt{3}$

So, the answer is $\sqrt{3}$.

To find $ an(frac{4pi}{3})$, locate the angle $frac{4pi}{3}$ on the unit circle. The angle $frac{4pi}{3}$ is in the third quadrant.

The reference angle for $frac{4pi}{3}$ is:

$360^circ – 240^circ = 120^circ – 60^circ = 60^circ$

or in radians:

$2pi – frac{4pi}{3} = frac{6pi}{3} – frac{4pi}{3} = frac{2pi}{3} = frac{pi}{3}$

The tangent of the reference angle is:

$ an(frac{pi}{3}) = sqrt{3}$

Therefore, $ an(frac{4pi}{3}) = sqrt{3}$.

To determine $ an(frac{4pi}{3})$, note that the angle $frac{4pi}{3}$ is in the third quadrant.

The reference angle is:

$frac{4pi}{3} – pi = frac{pi}{3}$

Since tangent in the third quadrant is positive, we get:

$ an(frac{4pi}{3}) = an(frac{pi}{3}) = sqrt{3}$

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