91Ó°ÊÓ

Using the fact that there are 12 vertices in a regular dodecagon (12-sided polygon), we can calculate the angle between two consecutive vertices in degrees using the formula: \(360^{\circ} / n\), where 'n' is the number of vertices. So, the angle between consecutive vertices for a dodecagon is: \(360^{\circ} / 12 = 30^{\circ}\) Now, convert this angle from degrees to radians, as we need radians for calculating trigonometric functions: \[\frac{30^{\circ} * \pi}{180^{\circ}} = \frac{\pi}{6}\] So, the angle between consecutive vertices in radians is \(\frac{\pi}{6}\).

The coordinates of the first vertex are given in the problem, which is (1,0).

To determine the coordinates of each vertex, we can use the following formula: \( x = cos(\theta), y = sin(\theta)\) Where (x, y) is the coordinate of the vertex, and \(\theta\) is the angle between consecutive vertices in radians. Starting from the angle of the first vertex, which is 0, we will proceed to find the coordinates of the other vertices counterclockwise: Vertex 1: (1,0), angle 0 Vertex 2: (cos(\(\frac{\pi}{6}\)), sin(\(\frac{\pi}{6}\))), angle \(\frac{\pi}{6}\) Vertex 3: (cos(\(2 \times \frac{\pi}{6}\)), sin(\(2 \times \frac{\pi}{6}\))), angle \(\frac{\pi}{3}\) Vertex 4: ..., angle \(\frac{\pi}{2}\) ... Vertex 12: ..., angle \(2 \times \pi - \frac{\pi}{6}\) Calculating each vertex coordinates: 1. (1, 0) 2. \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\) 3. \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) 4. (0, 1) 5. \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) 6. \(\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\) 7. \(-1, 0\) 8. \(\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\) 9. \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\) 10. (0, -1) 11. \(\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\) 12. \(\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\) So, the coordinates of all twelve vertices of the regular dodecagon on the unit circle, starting at (1,0) and going counterclockwise, are as follows: (1, 0); \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\); \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\); (0, 1); \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\); \(\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\); \(-1, 0\); \(\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\); \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\); (0, -1); \(\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\); \(\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)\).