1. **Problem Statement:** Determine the coordinates $(x_i, y_i)$ for $i=1$ to $5$ of five points equally spaced by angle on a quarter circle centered at the origin with radius 1, located at the bottom-left corner. 2. **Understanding the quarter circle:** The quarter circle is centered at the origin $(0,0)$ with radius $r=1$. Since it is at the bottom-left corner, the points lie in the third quadrant where both $x$ and $y$ are negative or zero. 3. **Angle spacing:** A quarter circle spans $90^\circ$ or $\frac{\pi}{2}$ radians. Five points equally spaced by angle means the angle between consecutive points is $\frac{\pi/2}{4} = \frac{\pi}{8}$ radians. 4. **Angles of points:** Starting from the negative x-axis (which is $\pi$ radians) and moving clockwise to the negative y-axis ($\frac{3\pi}{2}$ radians), the five points have angles: $$\theta_1 = \pi$$ $$\theta_2 = \pi + \frac{\pi}{8} = \frac{9\pi}{8}$$ $$\theta_3 = \pi + 2\cdot\frac{\pi}{8} = \frac{10\pi}{8} = \frac{5\pi}{4}$$ $$\theta_4 = \pi + 3\cdot\frac{\pi}{8} = \frac{11\pi}{8}$$ $$\theta_5 = \pi + 4\cdot\frac{\pi}{8} = \frac{12\pi}{8} = \frac{3\pi}{2}$$ 5. **Coordinates formula:** For a point on a circle of radius 1 at angle $\theta$, coordinates are: $$x = \cos(\theta)$$ $$y = \sin(\theta)$$ 6. **Calculate each point:** - $x_1 = \cos(\pi) = -1$, $y_1 = \sin(\pi) = 0$ - $x_2 = \cos(\frac{9\pi}{8}) = -\frac{\sqrt{2 - \sqrt{2}}}{2}$, $y_2 = \sin(\frac{9\pi}{8}) = -\frac{\sqrt{2 + \sqrt{2}}}{2}$ - $x_3 = \cos(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2}$, $y_3 = \sin(\frac{5\pi}{4}) = -\frac{\sqrt{2}}{2}$ - $x_4 = \cos(\frac{11\pi}{8}) = -\frac{\sqrt{2 + \sqrt{2}}}{2}$, $y_4 = \sin(\frac{11\pi}{8}) = -\frac{\sqrt{2 - \sqrt{2}}}{2}$ - $x_5 = \cos(\frac{3\pi}{2}) = 0$, $y_5 = \sin(\frac{3\pi}{2}) = -1$ 7. **Final answers:** $$x_1 = -1, \quad y_1 = 0$$ $$x_2 = -\frac{\sqrt{2 - \sqrt{2}}}{2}, \quad y_2 = -\frac{\sqrt{2 + \sqrt{2}}}{2}$$ $$x_3 = -\frac{\sqrt{2}}{2}, \quad y_3 = -\frac{\sqrt{2}}{2}$$ $$x_4 = -\frac{\sqrt{2 + \sqrt{2}}}{2}, \quad y_4 = -\frac{\sqrt{2 - \sqrt{2}}}{2}$$ $$x_5 = 0, \quad y_5 = -1$$