Identify the coordinates of the point on the unit circle at angle $ frac{7pi}{6} $

To find the coordinates of the point on the unit circle at angle $ \frac{7\pi}{6} $, use the unit circle values:

$ \frac{7\pi}{6} $

is in the third quadrant, where both sine and cosine are negative. The reference angle is $ \frac{\pi}{6} $, which corresponds to the coordinates:

$ (\cos(\pi/6), \sin(\pi/6)) = (\frac{\sqrt{3}}{2}, \frac{1}{2}) $

Since it is in the third quadrant, the coordinates are:

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

The final coordinates are:

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

To find the coordinates of $ frac{7pi}{6} $ on the unit circle, recognize it is in the third quadrant, where:

$ cos( heta) = -frac{sqrt{3}}{2} $

and

$ sin( heta) = -frac{1}{2} $

So,

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

In the third quadrant,

$ cos(frac{7pi}{6}) = -frac{sqrt{3}}{2}, sin(frac{7pi}{6}) = -frac{1}{2} $

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