91Ó°ÊÓ

We will begin by dividing the full circle into six equal parts, representing the central angles of the hexagon. Since there are 360 degrees in a full circle, and we want to find the coordinates of the vertices in counterclockwise order, we have to calculate the angles 0°, 60°, 120°, 180°, 240°, and 300°. Then, convert these degree measurements to radians: - 0° = 0 radians - 60° = \(\frac{\pi}{3}\) radians - 120° = \(\frac{2\pi}{3}\) radians - 180° = \(\pi\) radians - 240° = \(\frac{4\pi}{3}\) radians - 300° = \(\frac{5\pi}{3}\) radians

Now that we have the angle measures in radians, we can use them to calculate the coordinates of the vertices of the hexagon on the unit circle. We will apply the unit circle formula, given as \(P(\cos \theta, \sin \theta)\), where \(\theta\) is the angle in radians.

Plug the angle values in radians into the unit circle formula: 1. For \(\theta = 0\) radians: P(\(\cos 0, \sin 0\)) = P(1, 0) 2. For \(\theta = \frac{\pi}{3}\) radians: P(\(\cos \frac{\pi}{3}, \sin \frac{\pi}{3}\)) = P(\(\frac{1}{2}\), \(\frac{\sqrt{3}}{2}\)) 3. For \(\theta = \frac{2\pi}{3}\) radians: P(\(\cos \frac{2\pi}{3}, \sin \frac{2\pi}{3}\)) = P(\(-\frac{1}{2}\), \(\frac{\sqrt{3}}{2}\)) 4. For \(\theta = \pi\) radians: P(\(\cos \pi, \sin \pi\)) = P(-1, 0) 5. For \(\theta = \frac{4\pi}{3}\) radians: P(\(\cos \frac{4\pi}{3}, \sin \frac{4\pi}{3}\)) = P(\(-\frac{1}{2}\), -\(\frac{\sqrt{3}}{2}\)) 6. For \(\theta = \frac{5\pi}{3}\) radians: P(\(\cos \frac{5\pi}{3}, \sin \frac{5\pi}{3}\)) = P(\(\frac{1}{2}\), -\(\frac{\sqrt{3}}{2}\)) So the coordinates of the six vertices of the regular hexagon whose vertices are on the unit circle, in counterclockwise order starting at (1,0), are: (1,0), \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), (-1,0), \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\), and \(\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\).