Find the value of $ an left( frac{4pi}{3}
ight) $ on the unit circle

To find $ \tan \left( \frac{4\pi}{3} \right) $ on the unit circle, we note that $ \frac{4\pi}{3} $ radians is in the third quadrant.

In the third quadrant, both sine and cosine are negative. The reference angle for $ \frac{4\pi}{3} $ is $ \frac{\pi}{3} $.

We know that:

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

Since tangent is positive in the third quadrant:

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

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

To determine the value of $ an left( frac{4pi}{3}
ight) $, we know that $ frac{4pi}{3} $ places us in the third quadrant of the unit circle.

In this quadrant, tangent is positive. Considering the reference angle for $ frac{4pi}{3} $ is $ frac{pi}{3} $, we have:

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

We recall:

$ an left( pi + heta
ight) = an heta $

Thus:

$ an left( pi + frac{pi}{3}
ight) = an left( frac{pi}{3}
ight) = sqrt{3} $

We need to find $ an left( frac{4pi}{3}
ight) $. This angle lies in the third quadrant, where tangent is positive.

The reference angle is $ frac{pi}{3} $, and we know:

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

Thus:

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

Start Using PopAi Today