Find the exact coordinates of a point on the unit circle given that the point is $frac{7pi}{6}$ radians from the positive x-axis.

To determine the coordinates of the point on the unit circle at an angle of $\frac{7\pi}{6}$ radians, we use the sine and cosine functions:

The x-coordinate (cosine) is:

$\cos\left(\frac{7\pi}{6}\right) = \cos\left(\pi + \frac{\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}$

The y-coordinate (sine) is:

$\sin\left(\frac{7\pi}{6}\right) = \sin\left(\pi + \frac{\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2}$

Thus, the exact coordinates are:

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

Let’s find the coordinates of the given point using the unit circle properties.

For $ heta = frac{7pi}{6}$, we calculate:

$x = cosleft(frac{7pi}{6}
ight) = -cosleft(frac{pi}{6}
ight) = -frac{sqrt{3}}{2}$

$y = sinleft(frac{7pi}{6}
ight) = -sinleft(frac{pi}{6}
ight) = -frac{1}{2}$

The coordinates are:

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

The exact coordinates at $frac{7pi}{6}$ radians are:

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

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