$ ext{Find the sine, cosine, and tangent of } 150^{circ} ext{ using the unit circle}$

First, convert 150 degrees to radians:

$150^{\circ} = \frac{5\pi}{6} \text{ radians}$

Next, identify the coordinates of the corresponding point on the unit circle:

$\left(\cos\left(\frac{5\pi}{6}\right), \sin\left(\frac{5\pi}{6}\right)\right) = \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

Therefore,

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

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

To find the tangent:

$\tan(150^{\circ}) = \frac{\sin(150^{\circ})}{\cos(150^{\circ})} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

Convert 150 degrees to radians:

$150^{circ} = frac{5pi}{6} ext{ radians}$

The coordinates on the unit circle for this angle are:

$left(cosleft(frac{5pi}{6}
ight), sinleft(frac{5pi}{6}
ight)
ight) = left(-frac{sqrt{3}}{2}, frac{1}{2}
ight)$

So,

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

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

Finally,

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

Convert 150 degrees to radians:

$150^{circ} = frac{5pi}{6} ext{ radians}$

The coordinates on the unit circle are:

$left(cosleft(frac{5pi}{6}
ight), sinleft(frac{5pi}{6}
ight)
ight) = left(-frac{sqrt{3}}{2}, frac{1}{2}
ight)$

Thus,

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

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

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

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