Find the value of $ an( heta)$ on the unit circle where $ heta=150^circ$

To find $\tan(150^\circ)$, we first note that $150^\circ$ can be written as $180^\circ – 30^\circ$.

The reference angle here is $30^\circ$.

Since $\tan\theta$ is negative in the second quadrant:

$\tan(150^\circ) = -\tan(30^\circ)$

We know that $\tan(30^\circ) = \frac{1}{\sqrt{3}}\ or \ \frac{\sqrt{3}}{3}$.

Therefore,

$\tan(150^\circ) = -\frac{1}{\sqrt{3}}\ or \ -\frac{\sqrt{3}}{3}$

First, observe that $150^circ$ is in the second quadrant.

In the second quadrant, tangent is negative.

Also, $ an( heta) = frac{sin( heta)}{cos( heta)}$.

For $150^circ$, we have:

$sin(150^circ) = frac{1}{2}$

and

$cos(150^circ) = -frac{sqrt{3}}{2}$

Therefore,

$ an(150^circ) = frac{frac{1}{2}}{-frac{sqrt{3}}{2}} = -frac{1}{sqrt{3}} = -frac{sqrt{3}}{3}$

$ an(150^circ)$ can be calculated as:

$ an(150^circ) = – an(30^circ)$

Given $ an(30^circ) = frac{1}{sqrt{3}},$

$ an(150^circ) = -frac{1}{sqrt{3}} = -frac{sqrt{3}}{3}$

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