$ ext{Find the value of } an heta ext{ given that } heta ext{ is an angle on the unit circle with a terminal side passing through the point } left( -frac{1}{2}, -frac{sqrt{3}}{2}
ight).$

To find the value of $\tan \theta $, we use the fact that tan is defined as the ratio of the y-coordinate to the x-coordinate on the unit circle.

Given the point $\left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right)$, we have:

$\tan \theta = \frac{y}{x} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}$

Simplify the expression:

$\tan \theta = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{1} = \sqrt{3}$

Thus, the value of $\tan \theta$ is $\sqrt{3}$.

First, we identify the coordinates of the point given: $left( -frac{1}{2}, -frac{sqrt{3}}{2}
ight)$.

Tangent is the ratio of the y-coordinate to the x-coordinate:

$ an heta = frac{ ext{Opposite}}{ ext{Adjacent}} = frac{-frac{sqrt{3}}{2}}{-frac{1}{2}}$

Next, we simplify this ratio:

$ an heta = frac{-frac{sqrt{3}}{2}}{-frac{1}{2}} = frac{sqrt{3}}{1}$

The final value of $ an heta$ is:

$ an heta = sqrt{3}$

Given the point $left( -frac{1}{2}, -frac{sqrt{3}}{2}
ight)$:

$ an heta = frac{-frac{sqrt{3}}{2}}{-frac{1}{2}} = sqrt{3}$

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