Find the value of $ an(240)$ using the unit circle

To find the value of $\tan(240)$ using the unit circle, we first determine the corresponding point on the unit circle for an angle of 240 degrees.

240 degrees is in the third quadrant, where the tangent function is positive.

We can subtract 180 degrees to find the reference angle:

$240^{\circ} – 180^{\circ} = 60^{\circ}$

The reference angle is 60 degrees. The coordinates for 60 degrees on the unit circle are $\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)$.

Since we are in the third quadrant, both coordinates are negative:

$\left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right)$

The formula for tangent is:

$ \tan(\theta) = \frac{y}{x} $

Thus,

$ \tan(240^{\circ}) = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3} $

To find the value of $ an(240)$ using the unit circle, note that 240 degrees is in the third quadrant.

Subtract 180 degrees to get the reference angle:

$240^{circ} – 180^{circ} = 60^{circ}$

At 60 degrees, the coordinates are $left( frac{1}{2}, frac{sqrt{3}}{2}
ight)$. In the third quadrant, these become:

$left( -frac{1}{2}, -frac{sqrt{3}}{2}
ight)$

The tangent is:

$ an( heta) = frac{y}{x} $

Therefore:

$ an(240^{circ}) = frac{-frac{sqrt{3}}{2}}{-frac{1}{2}} = sqrt{3} $

$ an(240^{circ})$ is found in the third quadrant.

The reference angle is:

$60^{circ}$

The coordinates:

$left( -frac{1}{2}, -frac{sqrt{3}}{2}
ight)$

The tangent:

$ sqrt{3} $

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