Finding the Cartesian Coordinates from Polar Coordinates on a Unit Circle

Given a point on the unit circle with polar coordinates $(r, \theta)$, where $r = 1$ and $\theta = \frac{5\pi}{6}$, find the Cartesian coordinates $(x, y)$.

First, recall the conversion formulas from polar to Cartesian coordinates:

$ x = r \cos\theta $

$ y = r \sin\theta $

Since $r = 1$, substitute $\theta = \frac{5\pi}{6}$ into the formulas:

$ x = 1 \cdot \cos\left(\frac{5\pi}{6}\right) $

$ y = 1 \cdot \sin\left(\frac{5\pi}{6}\right) $

Using the unit circle values, we know:

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

$ \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} $

Thus, the Cartesian coordinates are:

$ x = -\frac{\sqrt{3}}{2} $

$ y = \frac{1}{2} $

Therefore, the Cartesian coordinates corresponding to the polar coordinates $(1, \frac{5\pi}{6})$ are:

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

To convert the polar coordinates $(1, frac{5pi}{6})$ to Cartesian coordinates, we use the equations:

$ x = r cos heta $

$ y = r sin heta $

Given $r = 1$ and $ heta = frac{5pi}{6}$, we find:

$ x = cosleft(frac{5pi}{6}
ight) $

$ y = sinleft(frac{5pi}{6}
ight) $

From the unit circle, we know:

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

$ sinleft(frac{5pi}{6}
ight) = frac{1}{2} $

Thus, the Cartesian coordinates are:

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

Given the polar coordinates $(1, frac{5pi}{6})$, we convert them using:

$ x = cosleft(frac{5pi}{6}
ight) $

$ y = sinleft(frac{5pi}{6}
ight) $

Thus:

$ x = -frac{sqrt{3}}{2} $

$ y = frac{1}{2} $

The Cartesian coordinates are:

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

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