Find the values of $sin$, $cos$, and $ an$ for an angle of 120 degrees using the unit circle.

To find the values of $\sin$, $\cos$, and $\tan$ for an angle of 120 degrees, first convert the angle to radians:

$120^\circ = \frac{120 \pi}{180} = \frac{2 \pi}{3}$

Next, locate the angle on the unit circle. The angle $\frac{2 \pi}{3}$ is in the second quadrant, where the sine is positive, and the cosine and tangent are negative.

The reference angle for $120^\circ$ is $180^\circ – 120^\circ = 60^\circ$.

For $60^\circ$, we have:

$\sin 60^\circ = \frac{\sqrt{3}}{2}$

$\cos 60^\circ = \frac{1}{2}$

Since 120 degrees is in the second quadrant:

$\sin 120^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2}$

$\cos 120^\circ = -\cos 60^\circ = -\frac{1}{2}$

$\tan 120^\circ = \frac{\sin 120^\circ}{\cos 120^\circ} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3}$

To determine the values of $sin$, $cos$, and $ an$ for $120^circ$, first convert the angle to radians:

$120^circ = frac{2pi}{3}$

Locate $frac{2pi}{3}$ on the unit circle, which is in the second quadrant. In the second quadrant, $sin$ is positive, and $cos$ and $ an$ are negative.

The reference angle is $60^circ$.

Using the reference angle $60^circ$:

$sin 60^circ = frac{sqrt{3}}{2}$

$cos 60^circ = frac{1}{2}$

Thus:

$sin 120^circ = frac{sqrt{3}}{2}$

$cos 120^circ = -frac{1}{2}$

$ an 120^circ = -sqrt{3}$

Convert $120^circ$ to radians:

$frac{2pi}{3}$

In the second quadrant, $sin$ is positive, $cos$ and $ an$ are negative. Reference angle is $60^circ$:

$sin 120^circ = frac{sqrt{3}}{2}$

$cos 120^circ = -frac{1}{2}$

$ an 120^circ = -sqrt{3}$

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